Financial activity based on natural peril events

ABSTRACT

A computer implemented method and system for automatically setting prices of financial products in a financial activity having a plurality of possible outcomes, includes receiving a first request from a participant terminal to purchase a financial product for one of the possible outcomes, i; and electronically computing a price for the requested financial product, in response to the first request, based at least in part on a first formula. Also included is a method for automatically updating the prices for all other outcomes, other than the outcome, i, based at least in part on a second formula. In one example, the financial products include contracts in a one-sided market of buyer participants where the outcomes are mutually exclusive and collectively exhaustive.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. non-provisionalpatent application Ser. No. 11/981,414, filed Oct. 31, 2007, now U.S.Pat. No. 7,783,542 which is a continuation-in-part of application Ser.No. 11/312,662, filed Dec. 20, 2005, now U.S. Pat. No. 7,693,766, and acontinuation-in-part of application Ser. No. 11/901,050, filed Sep. 14,2007, now U.S. Pat. No. 7,584,134, which is a continuation-in-part ofapplication Ser. No. 11/312,783, filed Dec. 20, 2005, now U.S. Pat. No.7,584,133.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention pertains to financial activities and in particularto such activities in which a financial return may be paid out based ona participant's prediction of weather-related or other naturallyoccurring events, and especially such events that are regarded as beingperilous.

2. Description of the Related Art

Oftentimes, natural peril events contain sufficient energy to imposepotentially significant financial burdens arising from damage toproperty. It is the nature of such catastrophes that they cannot bepredicted with exact certainty, even in severity or number ofoccurrences within an event season or the exact time and/or duration ofan event. These types of natural peril events include, for example,earthquakes, tornadoes and tropical weather events such as cyclones, (aterm given to all circulating weather systems over tropical waters—andof special interest here, the Atlantic basin and eastern Pacific).Tropical cyclones include tropical weather events referred to as“hurricanes” if they are sufficiently strong. Tropical cyclones whichgrow in intensity so as to become hurricanes originate at sea and maymake landfall and travel along a land portion before dissipating orreturning to the sea. Homeowners and business insurance policiestypically contain deductible provisions ranging from 2% to 15% of thevalue of a home or worksite.

Further, these same policies do not provide any coverage for the outsideareas of a home or business such as landscaping, outside lighting,docks, fencing and the like. Often, property owners do not havesufficient flood insurance or have other omissions or insufficientcoverage which result in catastrophic financial losses in even thelowest rated hurricanes. Great losses suffered by property owners, suchas those located along coastal and outlying areas, can be overwhelmingfor those who cannot afford to be self-insured. Insurance companiesoffer substantial aid for these individuals, but economic strains causedby unusually active hurricane seasons have resulted in relatively highpremiums. In order to make certain that insurance protection isavailable to individuals on an ongoing basis, various legislation andregulations have been enacted. However, substantial economic burdensremain, such as high deductible amounts and excluded items, whichrepresent damage costs which must be borne directly by the individual.Further, there are considerable delays in obtaining insurance relief,due to a number of factors outside of the owner's control, such asdelays associated with adjuster scheduling, claim processing andgovernmental determinations. These delays are considerably extended whenwide scale damage occurs.

As if the present problems are not enough, it has been predicted thatthe increased storm activity of the past few years is likely to continuein the Atlantic basin for the next 15 to 20 years. One prediction for2006 is that 17 named storms will occur, nine of which can becomehurricanes and five of which are expected to develop into major storms,with winds of 111 mph or more. By comparison, in the year 2005, 26 namedstorms were reported. Of the 13 major storms that formed the past twoyears, seven struck the U.S., whereas, according to the historicalaverage, only one of every three reported storms would be considered“major” storms.

In addition to increased whether severity, other factors have been citedas causes for unexpectedly large damage estimates. For example, it hasbeen estimated that, by year 2020, a single Miami storm could causecatastrophic losses of 500 Billion dollars—several times the damageinflicted by Hurricane Katrina. This is attributed to the rise inadditional property development demanded by a growing population, alongwith a rise in purchasing power with greater individual wealth. Theseestimates have not included any consideration of inflation.

Other lessons are being learned from hurricane Katrina. For example, theGreat Miami hurricane of 1926 caused about $760 million in damage, in2004 dollars. Surprisingly, if the hurricane were to be repeated at thepresent time (the same magnitude and following the same track) damage isestimated to be as large as $130 billion, due in large part topopulation expansion in the area. In the year 2020, damage estimatesfrom the same hurricane are estimated to be as great as $500 billion. Inaddition to primary damage factors such as loss of property, otherfactors directly result from a natural event. For example, the FederalEmergency Management Agency (FEMA) has encountered significantdifficulty in providing temporary housing for disaster victims. Loss ofdwellings is aggravated by extensive loss of jobs, further slowingeconomic and personal recovery. For example, FEMA's hotel program for2005 cost the federal government $325 million and, at its peak, coveredapproximately 85,000 rooms.

Other natural activities are also forecasted to exhibit alarming trends.For example, a tsunami occurring in the next few years is anticipated toresult in a 60 Billion dollar disaster, imperiling 1 million residents,affecting 600,000 jobs locally and 2.5 million jobs nationwide. TheCalifornia State Seismic Safety Commission and others have urged thatlocal evacuation plans be upgraded to account for possible catastrophiclosses. For example, in one instance, losses are estimated to includethe closure of the ports of Los Angeles and Long Beach. A two-monthshutdown of the ports would also cost $60 billion and affect 600,000jobs in the state and 2.5 million jobs nationwide.

SUMMARY OF THE INVENTION

The invention is generally directed to conducting financial activitiesbetween a provider and a plurality of participants, based on a naturalperil event such as a tropical cyclone, a tornado or an earthquake. Theparticipants are given an opportunity to predict the future occurrenceof the natural peril event, and to invest funds with the expectation ofa return on their investment if their prediction should match theoutcome of the natural peril event.

The invention in one implementation encompasses a computer implementedmethod and system for automatically setting prices of financial productsin a financial activity having a plurality of possible outcomes,includes receiving a first request from a participant terminal topurchase a financial product for one of the possible outcomes, i; andelectronically computing a price for the requested financial product, inresponse to the first request, based at least in part on a firstformula, whereinP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome i    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1

In another implementation, the invention encompasses a method forautomatically updating the prices for all other outcomes, other than theoutcome, i, based at least in part on a second formula, whereinP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing for outcome k.    -   Kπ^(k) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1

In another implementation, the invention encompasses financial productsthat include contracts in a one-sided market of buyer participants wherethe outcomes are mutually exclusive and collectively exhaustive.

In yet another implementation, the invention encompasses a computerimplemented method for automatically pricing financial products in afinancial activity for a plurality of possible outcomes, consistent withthe risks as perceived at any given time by the activity participants,comprising the steps of:

receiving a first request from a participant terminal to purchase afinancial product for one of the possible outcomes, i;

electronically computing a price for the one possible outcome based atleast in part on the following first formulaP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome I,    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1

updating all other prices based at least in part on the following secondformulaP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing probability for        outcome k.    -   Kπ^(k) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1.

In a further implementation, the present invention is directed to acomputer implemented system for automatically setting prices offinancial products in a financial activity having a plurality ofpossible outcomes, comprising:

a network for receiving a first request from a participant terminal topurchase a financial product for one of the possible outcomes, i;

a computer having memory with a data structure stored in said memory,said data structure comprising the following first formulaP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome I,    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically computes a price for the requested        financial product, in response to the first request, based at        least in part on the first formula.

If desired, the data structure could further comprise the followingsecond formulaP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing probability for        outcome k.    -   Kπ^(k) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically automatically updates the prices for        all other outcomes, other than the outcome, i, based at least in        part on the second formula

In another implementation, the present invention is directed to anarticle of manufacture including a machine readable medium for causing acomputer system to automatically set prices of financial products in afinancial activity having a plurality of possible outcomes, comprising:

an input for receiving a first request from a participant terminal topurchase a financial product for one of the possible outcomes, i; and

a data structure comprising the following first formulaP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome I,    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically computes a price for the requested        financial product, in response to the first request, based at        least in part on the first formula.

If desired, the machine readable medium could further comprise thefollowing second formulaP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing probability for        outcome k.    -   Kπ^(k) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically automatically updates the prices for        all other outcomes, other than the outcome, i, based at least in        part on the second formula

In another implementation, the present invention can be employed as aviable solution to the economic and financial devastation which affectscitizens as well as governments caused by naturally occurringcatastrophes such as hurricanes. The present invention can be employedto offer an economic solution which does not use government or publicfunds and therefore does not require tax payer funding to replenishgovernment pools. In one aspect, the present invention can be employedto use only funds provided by private entities to augment governmentalfinancial assistance for catastrophic occurrences.

BRIEF DESCRIPTION OF THE DRAWINGS

Features of exemplary implementations of the invention will becomeapparent from the description, the claims, and the accompanying drawingsin which:

FIG. 1 is a schematic representation of a financial activity networkaccording to an embodiment of the present invention;

FIG. 2 is a schematic representation of a financial activity apparatusimplementing the present invention;

FIG. 3 is a schematic representation of a first participant terminal;

FIG. 4 is a schematic representation of a second participant terminal;

FIG. 5 is a schematic representation of a point-of-purchase participantterminal;

FIG. 6 is a schematic representation of a standalone participantterminal;

FIG. 7 is a schematic representation of a first data display;

FIGS. 8 a-8 f are schematic representations of a series of screendisplays; and

FIGS. 9-12 together comprise a schematic flow diagram representing oneexample of system operation.

FIGS. 13-18 are graphical depictions of data screens implementing afinancial activity;

FIG. 19 is a schematic drawing illustrating treatment given to a unitarea addressed in an exemplary financial activity;

FIG. 20 is a schematic representation of the relationship betweenfinancial investment unit prices and dates of participation in afinancial activity, prior to occurrence of a natural peril event;

FIG. 21 is a table showing examples of illustrative financial investmentunit prices;

FIG. 22 is a table showing Poisson probabilities and otherprobabilities;

FIG. 23 is a table illustrating trade-offs involved in choosing thenumber of financial activities to be run.

FIG. 24 is a schematic representation of a computer screen showing anintroduction and overview presented to a user;

FIG. 25 is a schematic representation of a computer screen showing anoverview of a graphical user interface reflecting current conditions;

FIG. 26 is a schematic representation of a first computer screen showingthe graphical user interface directed to a detailed geographicallocation;

FIG. 27 is a schematic representation of a second computer screenshowing the graphical user interface directed to a detailed geographicallocation;

FIG. 28 is a schematic representation of a computer screen showing thegraphical user interface directed to an overview of geographicallyrelated data for a given year;

FIG. 29 is a schematic representation of a computer screen showing thegraphical user interface offering menu choices to a user, within thecontext of the chosen year;

FIG. 30 is a schematic representation of a computer screen showing thegraphical user interface offering menu choices to a user, within thecontext of a given storm occurring in the chosen year;

FIG. 31 is a schematic representation of a computer screen showing thegraphical user interface offering different sets of data to a user, thatare available within the context of a given storm, occurring in thechosen year;

FIG. 32 is a schematic representation of a computer screen showing thegraphical user interface identifying a first chosen set of data selectedby a user, for a particular day and time, that is available within thecontext of a given storm occurring in the chosen year;

FIG. 33 is a schematic representation of a computer screen showing thegraphical user interface identifying a data event selected by a user,for a first particular geographical position for a particular day andtime; that is available within the context of a given storm occurring inthe chosen year;

FIG. 34 is a schematic representation of a computer screen showing thegraphical user interface identifying a data event selected by a user,for a second particular geographical position for a particular day andtime, that is available within the context of a given storm occurring inthe chosen year;

FIG. 35 is a schematic representation of a computer screen showing thegraphical user interface displaying another menu choice chosen by auser, within the context of a given storm occurring in the chosen year;

FIG. 36 is a schematic representation of a computer screen showing thegraphical user interface with sets of probability data available forselection by a user, for a particular day and time, that is availablewithin the context of a given storm occurring in the chosen year;

FIG. 37 is a schematic representation of a computer screen showing thegraphical user interface with probability data selected by a user, for aparticular geographic location, day and time, that is available withinthe context of a given storm occurring in the chosen year;

FIG. 38 is a schematic representation of a computer screen showing thegraphical user interface identifying a forecast data event selected by auser, for a particular geographical position for a particular day andtime, that is available within the context of a given storm occurring inthe chosen year;

FIG. 39 is a schematic representation of a computer screen showing thegraphical user interface with probability data for multiple naturalperil events of a given year, along with related investment data;

FIG. 40 is a schematic representation of a computer screen showing thegraphical user interface displaying investment data for a particulargeographical position for a particular day and time;

FIG. 41 shows a display table of FIG. 40;

FIG. 42 shows a series of display tables;

FIG. 43 shows candidate pricing curve functions for an exemplary study;

FIG. 44 shows an exemplary time series of prices for an exemplary study;

FIGS. 45( a) and 45(b) show average payouts and price volatility for aparticular study;

FIG. 46 shows a table of data for a pari-mutuel, or mutual risk pool;

FIGS. 47 and 48 show a table of data for a pari-mutuel market; and

FIG. 49 shows a schematic representation of a financial activityaccording to principles of the present invention; and

FIG. 50 is a schematic diagram illustrating performance of an adaptivecontrol algorithm with respect to market participants' beliefs about theoutcome probabilities.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The invention disclosed herein is, of course, susceptible of embodimentin many different forms. Shown in the drawings and described hereinbelow in detail are preferred embodiments of the invention. It is to beunderstood, however, that the present disclosure is an exemplificationof the principles of the invention and does not limit the invention tothe illustrated embodiments.

For ease of description, financial activity networks and other systems,articles of manufacture, and apparatus embodying the present inventionare described herein below in their usual assembled position as shown inthe accompanying drawings and terms such as upper, lower, horizontal,longitudinal, etc., may be used herein with reference to this usualposition. However, the systems, articles of manufacture and apparatusmay be manufactured, transported, sold, or used in orientations otherthan that described and shown herein.

I. Introduction

On-line performances have been proposed for a variety of financialactivities. These activities may be divided in a number of differentways such as gambling and non-gambling activities, for example. Gamingor gambling systems are in place which emulates traditional gamblingactivities in an on-line, internet software-based program, giving theusers the look and feel of traditional gambling activities. Suchgambling variants have been extended to nontraditional games of chancebased on virtually any experience known to mankind. Although gamingactivities can receive substantial benefit from implementations of thepresent invention, the present invention can also be employed with gamesof skill and non-gaming financial transactions.

Financial activities may also take the form of sweepstakes offerings inwhich participants are eligible to participate in a drawing or otherselection of winners who share in or are otherwise are entitled to apool of money set aside for the purpose. In another example of financialactivities contemplated by the present invention, cell phone subscriberscan text message their vote, indication of approval or choice ofalternatives and be entered into a sweepstakes for their participation.

In one instance, a financial activity is concerned with financialactivity based upon games of skill in which participants processhistoric and other data and apply scientific principles and acquiredanalytical skills to arrive at informed decisions relating to thepredictions of future naturally occurring activities. In one instance, aparticipant makes an investment based upon one or more informeddecisions, thereby contributing to a common pool from which payouts aremade, for example, based upon the accuracy, level of detail and timingof their prediction.

The present invention, in one instance, is also concerned withtraditional financial activities which lie outside the area of gambling.More specifically, in one example, the present invention is concernedwith allocating and managing a pool of money collected fromparticipants, i.e. market participants who seek to offset anunpredictable but nonetheless potentially catastrophic financial burdencaused by property loss due to natural peril events. In one instance,these types of financial activities are based upon the participant'sproperty interests. Although such financial investments do at times comewith a substantial element of risk in the expectation and amount ofreturn, it has long been recognized that such financial investments donot represent gambling activities.

The present invention finds immediate application in the field ofoffsetting losses of those who live or find themselves in a particulargeographic area subjected to natural peril events which causesignificant damage to their property interests (e.g. homes, businesses,rentals, and vacation shares). In one instance, the present invention isconcerned with offsetting losses without governmental interaction.Examples of such natural peril events supported by the present inventioninclude naturally occurring phenomena without human causality, such astropical weather systems such as hurricanes as well as other tropicalweather events.

The present invention is directed not only to catastrophic or disastertype weather or other natural events but also to “regular,”non-catastrophic or non-disaster weather or other natural events such asrain, snow, heat and cold usually regarded as severe or unwelcome, butnot catastrophic. For example, the present invention is directed tonatural events of virtually any type which are known to place financialor other burdens on people who suffer as a result of the event. in oneaspect, the financial activity contemplated herein need not be based onseverity of the natural event. Examples include business and/or personalloss (vacations, etc.) that could be hedged against the risk of naturalweather events such as (but not limited to) extreme days of rain, snow,heat, and cold. Excess cold and/or rain can disrupt the tourism andvacation industry. Unexpected snowfall can also disrupt travel and/oroutdoor events and lack of snowfall and high temperatures could disruptthe ski industry. Too many days of any weather extreme could affectbusinesses. This model provides ways of hedging against any of theseevents should a minimum threshold be exceeded (example, more than 5 raindays during a specified number of period in a specific geographic areaduring a certain time of the year) or could even compensate individualsand/or businesses on an escalating scale (binary) based upon the numberof rain days exceeded in a given period for a given geographic area andtime of the year. Based upon the minimum conditional payout guarantees,one could lock in a minimum payment based upon a threshold being met.

The premise of the financial activity is that market participants canexercise personal initiative to research known naturally occurringactivity and make certain predictions relating to forthcoming activityfor a particular season or year. A return on their investment can bepaid out at season's end or once a natural peril event has concluded,with the return being based upon certain factors such as the accuracy oftheir prediction, the amount of time between their investment and theoccurrence of a natural peril event, and the skillful use of establishedlaws of statistics and other resources available to a marketparticipant. Time and effort spent in researching natural activity canhelp to improve the accuracy of predictions made.

Any of the financial activities herein may be undertaken with one ormore participants, and the provider may comprise one or more separateentities.

II. External Objective Information Sources

In one instance, the financial activity incorporates conclusions;findings and reports of one or more external objective informationsources such as an independent disinterested third party that providespublicly available information and conclusions. The terms “external” and“independent” refer to entities which are separate from the providerand/or participants of the financial activity. In one instance, anexternal objective information source provides information pertaining toa natural peril event (in one example an indication that a definednatural peril event has occurred, and in another example, that thedefined natural peril event has concluded). In another instance, theexternal objective information source provides natural peril event data,such as a point of landfall or a land track of a storm originating atsea.

As discussed herein, the present invention is directed to financialactivities between a provider and one or more participants. Activitiesinvolving multiple participants are attractive in many instances, sincethe risk to a provider can be spread across a number of participants.However, the present invention also contemplates activity of virtuallyany type herein, conducted with a single participant who wishes tomitigate the risk of suffering setbacks due to a future natural event.The provider, must however, be willing to engage such an activity.

In one instance, the financial activities incorporate independentobjective information which is based upon naturally occurring eventsthat are studied by the external objective information source (andwhich, in one instance, is publicly available). In one example,financial activities contemplated by the present invention are directedto predictions of natural peril event activity, with financialinvestments being made before the activity has occurred and/orconcluded. Payouts or the assigning of points or other value accordingto a successful, matching outcome of the financial activity may be basedupon multiple factors, some of which are determinable by an externalobjective information source. Examples include the measurable severitywhich can be measured, for example, by the amount of property damage asdetermined by insurance institutes and government agencies. Payouts canalso depend upon metrics associated with the natural peril event, basedon details concerning occurrence of the natural peril event, such as itspoints or path of terrestrial contact and the timing of a participant'sinvestment such as the amount of lead time given by the marketparticipant prior to event conclusion.

In many instances, an external objective independent information sourceis relied upon for its expertise in studying, and measuring naturalperil event phenomena as well as drawing conclusions from data collectedfrom natural peril events. Oftentimes, reports made available to thepublic and others include inferences and conclusions drawn by an expertagency, which goes beyond a mere relating of observed facts. Informationwhich is key to carrying out a financial activity (such as the time andplace of occurrence of the natural activity) is obtained from a sourceexternal to the provider of the financial activity.

As will be seen herein, it is generally preferred that the identity ofthe external objective information source is defined beforehand, in arules database or in some other manner. In one instance, the externalobjective independent information source provides facts and conclusionswhich are made generally available to the public, or at least toindividuals likely to participate in the financial activity. In thismanner, individuals interested in participating in the financialactivity and those participants already engaged in the financialactivity can monitor progress of a natural peril event, independent ofthe financial activity. In one instance, participants in the financialactivity will be able to readily obtain expert information andskill-building technical intelligence from sources independent of thefinancial activity, thus enhancing motivation of the participants toengage in the financial activity with a greater likelihood of enjoying asuccessful outcome. Of particular interest here, are property ownersunable to obtain adequate insurance, but who nonetheless live in an areaknown to be subject to destructive natural peril events. Such propertyowners will have an interest in gaining an ability to predict naturalperil events, so as to be better able to protect their propertyinterests and to offset unforeseen damages. Others that might beinterested are cities and municipal corporations. Besides those thathave property interests at risk, interested parties may include thoseinterested in speculation types of financial activities, such asoperators of hedge funds and institutional type market participants.

Examples of external objective independent information sources includethe National Hurricane Center and the Tropical Prediction Center forhurricanes and other tropical storms.

Low-pressure tropical weather systems, or storms beginning at sea, startas relatively low energy thunderstorms. If a moving area ofthunderstorms in the tropics maintains its identity for 24 hours ormore, the weather system is termed a “tropical disturbance”. If theweather system exhibits a rotating or circulating weather pattern, theweather system is referred to as a tropical cyclone. The lowest energytropical cyclone is termed a “tropical depression” if its maximumsustained wind speed does not exceed 38 mph. For maximum sustained windspeeds ranging between 39 and 73 mph, the weather system is termed a“tropical storm”. The most intense weather systems are referred toeither as “hurricanes” or “typhoons”.

According to the National Hurricane Center, “hurricane” is a name for atropical cyclone that occurs in that oceanic area generally referred toas the Atlantic Basin or the eastern Pacific, and which is defined bymaximum sustained wind speeds of 74 mph or more. “Tropical cyclone” isthe generic term used for low-pressure systems exhibiting rotationalcharacteristics that develop in the tropics and meet a criterion forrelatively high maximum sustained wind speeds. The intensity ofhurricanes is measured according to the Saffir-Simpson scale (categories1 to 5).

It is well known that hurricanes and lesser storms develop from tropicaldepressions in the oceans where weather related factors cooperate toform and contribute energy to a low-pressure weather system. The weathersystem then travels across the ocean, along an unpredictable path. Ofinterest here are weather systems exhibiting circulating behavior whichgrow in intensity so as to develop into hurricanes which travel with agenerally westward direction component and make landfall or otherwisecontact property along well-defined geographical areas, such as theeastern coast of Canada, the United States, Mexico, and South America aswell as sea islands lying in a path of travel toward those land bodies.The class of storms referred to as “hurricanes” vary in intensity andare generally free to travel along their own unique pathway or “track”.Hurricanes are observed by independent expert agencies of the UnitedStates government, such as the National Hurricane Center (an example ofan external objective information source), which carefully records,analyzes and later publishes reports, findings and conclusions, whichare made available to the public.

Hurricanes are powered by the sea's thermal energy and by energy in theatmosphere. Generally speaking, hurricanes are directed by the easterlytrade winds. Around the center, core or “eye,” wind speeds accelerate togreat velocities. Moving ashore, these energetic winds displace theocean inwardly, toward land and are known to spawn tornadoes and producetorrential rains and floods. In the Atlantic Basin, for example,statistically there is an annual average of 8.6 tropical storms for theyears 1886-1998. Of these, 5.0, statistically, are hurricanes. The aboveillustrates aspects of natural peril events which may be employed in arules database or program or other structure to govern operation offinancial activity.

It is important to define early on those natural peril events which willqualify for consideration by the financial activity. For example,hurricanes considered by the financial activity may be limited to onlythose hurricanes which make terrestrial contact or which have a minimumstrength. In another example, it is important to define natural perilevents considered by the financial activity when the members of thepublic may have alternative definitions which vary from those to beconsidered by the financial activity. For example, participants whosuffered damage during the time of a tropical cyclone may not fallwithin the “best-track” or other report issued by an external,independent, objective information source (such as NHC/TPC) (informationemployed according to one of the possible rules of operation). Contraryto popular expectations, a particular participant may suffer damage fromnatural phenomena lying outside of the tropical event of interest to thefinancial activity.

Other examples of external objective independent information sourcesinclude the National Hurricane Center for hurricanes and other tropicalstorms, the National Weather Service and the Storm Prediction Center fortornadoes in the United States, and the Geological Survey, NationalEarthquake Information Center for earthquakes in the United States.Additional examples include third parties that provide indexes of damageor risk of damage for financial investors and insurance companies.

III. Examples of Natural Peril Events

Consideration will now be given to a few examples of independentlyobservable, moving or otherwise changing natural peril events, and inparticular, those events capable of causing substantial damage.

A tornado is a violently rotating column of air in contact with theground that extends from a particular type of cloud formation referredto as a “cumulonimbus cloud”. Tornadoes can be categorized as “weak”,“strong”, and “violent”; with weak tornadoes often having a thin,rope-like appearance, with rotating wind speeds no greater than about110 MPH. The typical “strong” tornado often has a recognizablefunnel-shaped cloud associated with a high energy whirling updraft androtating wind speeds ranging between 110 and 200 MPH. Tornado severityis also measured according to the Fujita scale.

Scientific tornado research is performed by the National Severe StormsLaboratory (NSSL) which has been a leader in Doppler radar development,research and testing, and which has run numerous field programs to studytornadoes and other severe weather. Tornado research is also conductedby NCAR (National Center for Atmospheric Research) located in Boulder,Colo., and TORRO located in the United Kingdom. Almost every universitywith an atmospheric science program, as well as many local NationalWeather Service (NWS) offices, have also published some tornado-relatedstudies. The National Weather Service is a branch of the United Statesgovernment organized under the US Dept of Commerce, and the NationalOceanic and Atmospheric Administration (NOAA). In the United States, theNational Weather Service (NWS) issues tornado forecasts nationwide.Warnings come from each local NWS office. The Storm Prediction Center,located in Norman, Okla. and organized under NOAA/National WeatherService, and the National Centers for Environmental Prediction, issuewatches and severe weather outlooks. Information concerning tornadoes inCanada is managed by the Meteorological Service of Canada.

Tropical cyclones are low-pressure weather systems that develop at sea,especially at low latitudes, usually beginning as relatively low-energytropical depressions. As storm energy builds, tropical depressions beginto exhibit a rotating or circular weather pattern, and if the stormintensity is sufficient it is classified as a tropical cyclone. Tropicalcyclones include “tropical storms,” but the most intense tropicalcyclones are referred to (depending on the ocean basin in which theyoccur) as “hurricanes,” “typhoons,” or simply “cyclones.” According tothe National Hurricane Center, “hurricane” is a name for a tropicalcyclone that occurs in that oceanic area generally referred to as theAtlantic Basin, and which is defined by certain minimum wind speeds.“Tropical cyclone” is the generic term used for low-pressure systems ofgreat intensity that develop in the tropics and meet a criterion forrelatively high minimum sustained wind speeds. The intensity ofhurricanes is measured according to the Saffir-Simpson scale.

It is well known that hurricanes and less energetic tropical weatherevents develop from tropical depressions in the oceans, where weatherrelated factors cooperate to form and contribute energy to alow-pressure weather system. The weather system then travels across theocean, along an unpredictable path. Of interest here are weather systemsexhibiting circulating behavior which grow in intensity so as to developinto hurricanes which travel with a generally westward directioncomponent and make landfall or otherwise contact property alongwell-defined geographical areas, such as the eastern coast of Canada,the United States, Mexico, South America and sea islands lying in a pathof travel toward those land bodies. The class of storms referred to ashurricanes vary in intensity and are generally free to travel alongtheir own unique pathway or “track”. Hurricanes are observed byindependent expert agencies of the United States government, such as theNational Hurricane Center (an example of an external objectiveinformation source), which records, analyzes and later publishesreports, findings and conclusions, which are made available to thepublic.

An earthquake is a shaking of the ground caused by the sudden breakingand shifting of large sections of the earth's rocky outer shell. Theearth is divided into three main layers—a hard outer crust, a softmiddle layer and a center core. The outer crust is broken into massive,irregular pieces called “plates.” An earthquake is associated with asudden motion or trembling in the Earth caused by the abrupt release ofstrain slowly accumulated in the Earth's plates. It is reasonable toexpect that future earthquakes in known earthquake areas will havemagnitudes generally comparable to the magnitudes of past earthquakes.Seismologists use a magnitude scale, such as the Richter scale, toexpress the seismic energy released by an earthquake.

Typical effects of earthquakes vary from a Richter magnitude less than3.5 (generally not felt, but nonetheless recorded) to a Richtermagnitude 8 or greater (so-called “great earthquakes” which can causeserious damage over large areas measured in hundreds of kilometers).Although earthquakes can have similar magnitudes, their effects can varygreatly according to distance, ground conditions, and man-madeconstruction techniques. Publicly available, independent third-partystudies are reported by the U.S. Geological Survey. For example, the USgeological survey reports Earthquake Density Maps of the United States,as well as Shake Maps and Seismogram Displays for recent earthquakeactivity in United States and throughout the world. The U.S. GeologicalSurvey, National Earthquake Information Center, World Data Center forSeismology in Denver, issues detailed reports of significant earthquakeactivity for local areas of the United States and maintains anearthquake catalog useful for earthquake predictions.

The above illustrates aspects of exemplary natural peril events whichmay be employed in a rules database, program, derivatives contract orother structure to govern operation of financial activity. It isimportant to define early on those natural peril events which willqualify for consideration by the financial activity. For example,tornadoes considered by the financial activity may be limited to onlythose tornadoes which make terrestrial contact or which have a minimumstrength. In another example, it is important to define natural perilevents considered by the financial activity when the members of thepublic may have alternative definitions which vary from those to beconsidered by the financial activity. For example, participants whosuffered damage during the time of a tropical cyclone may not fallwithin the “best-track” issued by an external, independent, objectiveinformation source (such as NHC/TPC) set up by the financial activity asthe conclusive source of such information. That is, contrary to popularexpectations, a particular participant may suffer damage from naturalphenomena which are not encompassed by the financial activity.

As mentioned, the present invention contemplates natural peril eventsthat are not related to hurricanes. Such natural peril events may bedescribed as natural peril events originating either on land (such astornadoes, hail storms, wildfires or earthquakes), at water level (suchas tsunamis and rogue waves caused by winds in contact with the watersurface) or below water level (such as tsunamis caused by underwaterearthquakes).

IV. Examples of Financial Activity Models

Different types of financial activities are carried out, typicallybetween a financial activity provider and one or more participants inthe financial activity. The activity may also be conducted betweenpaired or matched trading partners, as where bids are matched withpurchases. The pools of money supporting the financial activity can bemanaged as a mutual risk pool or not. For example, a financial activitycan be conducted to provide a fixed payout to one or more eligibleparties. Pricing can be structured in a number of different ways,preferably with prices determined by market conditions, and morepreferably by one or more algorithms. Different models of financialactivities are contemplated, including:

1. Derivative trading type of financial activity, such as thoseactivities directed to derivative securities interests, which aretypically monitored by the Securities and Exchange Commission or otheroversight bodies, such as the Commodity Futures Trading Commission (anindependent agency of the United States government), the New York StockExchange, the Chicago Mercantile Exchange, the Iowa Electronic Market,and others. Examples include futures contracts, options contracts andoptions on future contracts. Trading may be unilateral or multilateraland may be cleared, either initially or later on, by an exchange.

2. Secondary trading of financial assets developed by a participant ofone or more of the financial activities indicated herein, particularlysecondary trading between a participant of an ongoing financial activityand a third party wishing to deal directly with the participant, ratherthan the financial activity provider. The financial activity providermay require registration of the secondary trade or impose other controlsover the parties involved, including assistance with executing the tradebetween two or more participants or nonparticipants, such asregistration of the instruments traded.

3. Persons with cell phones or other portable communication devices suchas pda's or laptop computers, may vote their choice of first landfalland are charged for placing the vote, with price setting according toprinciples of the present invention. Voting may be carried out via textmessaging, the internet or voice communication, with or without the useof voice recognition technology. Those participating may, for example,be entered into a sweepstakes that may or may not include a secondsweepstakes for those who vote in favor of a particular choice.

4. Price-oriented competition, preferably in games of skill whereparticipants are charged an “entry fee” to engage in skillfulcompetition with other participants, with price setting according toprinciples of the present invention. The distribution or “prize” toqualifying participants is predetermined at the outset of competition,and accordingly is not affected by variability factors. However, ifdesired, the “entry fee” can be adjusted by variability factors such asthose relating to the time interval between investment and eventoutcome, and the probability of a successful outcome determinedapproximately at the time of investment.

5. Property value protection, particularly financial activities in whichthe participants are screened for eligibility to engage in the financialactivity, depending upon some indication of their property rights in ageographical area covered by the financial activity. Prices are setaccording to principles of the present invention

6. Games of skill, preferably where the participants are obliged todemonstrate a level of skill which pertains to the natural peril eventsof interest. Prices are set according to principles of the presentinvention

V. Financial Activity Network

Referring now to the drawings, and initially to FIGS. 1-4, financialactivity system 100 in one instance, takes the form of a financialactivity network generally indicated at 10. In one example, network 10includes a central managing system 12 linked to a plurality ofparticipant terminals 14-18. The terminals can comprise, for examplevirtually any device that provides communication with a workstation suchas a network or other computers including desktop, portable, lap-top ormainframe computers, data terminals, dumb terminals, personal digitalassistants, cellular phones or other electronic devices havingcommunications capabilities. Throughout the description given herein,“computer” refers to a computer system comprising one or more computingdevices, but could also refer to computer systems operated by stockmarkets, options exchanges, commodity exchanges or the like.

As schematically indicated in FIG. 1, each of the participant terminalscommunicate directly with central managing system 12 via communicationnetwork components 15, allowing concurrent transactions and datatransfers to occur. Other types of arrangements are possible. Forexample, communications between the central managing system and theparticipant terminals can employ virtually any communications technologyknown today. The geographical spacing between the central managingsystem and the participant terminals can have virtually any scaledesired. For example, the entire network 10 can be located in a singleroom, or in a single building or building complex or campus.

As will be seen herein, financial activity can take place according todifferent models. One model is directed to a derivative tradings type offinancial activity, such as those activities directed to derivativesecurities interests. In this type of financial activity, the centralmanaging system 12 preferably comprises a brokerage system communicatingwith an exchange system. Preferably, participants' trading is conductedthrough the brokerage system before being conducted with the exchange.If desired, in this type of financial activity, the brokerage system canbe omitted with participants dealing directly with the exchange system.If desired, the central managing system can either be incorporated into,or be provided in addition to, the exchange system.

Alternatively, the financial activity network 10 can be located atvarious nationwide or international locations, as may be desired. As afurther alternative, the financial activity network may take on anyform, and may employ wire, cable or wireless components (such as cellphones, text messaging devices, pda's, etc), for example. Network 10 canbe configured as an open connection or network such as the Internetnetwork, a wide area network, a telephone network, a satellite network,an on-line network or a closed circuit television network or the likeintra-facility network. Network 10 can also take the form of an Ethernetarrangement, a token ring, a token bus or any other suitablecommunications arrangement or configuration that can link workstations,particularly workstations including one or more data processingcomputers.

Financial activity can, but need not necessarily, take place eitherwithin or across local, state, federal, national or internationalboundaries. For example, participant terminals can be located in one ormore boundaries, e.g., political boundaries different from that of thecentral managing system. As a further example, although the centralmanagement system and the participant terminals are located within agiven boundary, e.g., a given political boundary, the central managementsystem may communicate with external objective, independent informationsources, external credit agencies or other agencies located within oneor more different political or geographic boundaries. As will be seen,operators of the financial activity may rely entirely on outsideservices to provide the needed credit and other financial arrangements.

The terminals can comprise one of the many different types of electronicdevices known today, including a programmable computer, a telephone, astand-alone machine such as an ATM machine, a television or a set-topbox unit, a credit card reader, a kiosk terminal, a point-of-purchaseregister, or a stand-alone unit resembling an arcade game, for example.The terminals preferably include an input device suitable for receivinga purchase request or other data from a participant, such as thoseemployed by purchasers to obtain goods and/or services from a merchant.The input devices can take many forms, including a keypad (includingthose used in cellular and other portable devices), keyboard touchscreen or mouse or a remote control device, contactless payment system,or fingerprint or other biometric system, for example. Systems, articlesand apparatus preferably comprise digital devices, but could alsocomprise analog or hybrid electronic or non-electronic devices, as maybe desired.

VI. System Apparatus

Turning to FIG. 2, financial activity system 100 includes systemapparatus 13 embodying the central managing system 12 shown in FIG. 1.In one example, system apparatus 13 comprises one or more storagedevices 20, one or more processors 22, and one or more interfacecomponents 24. The processor 22 in one example comprises a centralprocessor unit (“CPU”). The processor 22 executes one or moreinstructions of one or more programs 30, under control of an operatingsystem 35 employing one or more system programs 37. The program 30 inone example comprises one or more subroutines and one or more variables,as will be understood by those skilled in the art. The storage device 20in one example comprises at least one instance of a recordable datastorage medium, as described herein. The storage device 20 stores theprogram 30, and one or more databases 32, and one or more data files 34.

The interface component 24 in one instance comprises a graphical userinterface (“GUI”). In one example, the interface component 24 allows aservice provider or other user 38 to execute one or more programs 30.The program 30, in one example, comprises one or more subroutines, tocarry out the financial activity methods and operations to be describedherein. In one instance, program 30 includes one or more subroutines tocollect, publish, interpret or otherwise process information whichsupports principles and other aspects of operation of the financialactivity. In another instance, program 30 includes one or moresubroutines for implementing rules of operation for the financialactivity.

In another example, the interface component 24 allows the user 38 toverify or otherwise interact with one or more results of the program 30.In yet another example, the interface component 24 allows the user 38 toset one or more input values or operating parameters for the program 30.In the preferred embodiment illustrated in FIG. 2, interface component24 includes a display device 42 and a data input device 44 which allow auser 38 to set up the central managing system according to desiredoperating objectives. With the interface component 24, a user canaccess, read or write to the program 30, the databases 32 and the datafiles 34

Included in the apparatus 13 embodying system 12 is a communication port50 which provides two-way communication with the terminals 14-18.Communication port 50 can employ virtually any communications protocol,data format and other organizational, communication or other knowncontent that is in use today. It is generally preferred that thecommunications network employed between the central managing system andthe participant terminals comprise an interactive device taking anysuitable form.

The financial activity system 100, in one example, comprises a pluralityof components such as one or more of electronic components, hardwarecomponents, and computer software components. A number of suchcomponents can be combined or divided in the financial activity system100. An exemplary component of the financial activity system 100 employsand/or comprises a set and/or events of computer instructions written inor implemented with any of a number of programming languages, as will beappreciated by those skilled in the art. The financial activity system100 in one example comprises any (e.g., horizontal, oblique, orvertical) orientation, with the description and figures hereinillustrating one exemplary orientation of the financial activity system100, for explanatory purposes.

The financial activity system 100 in one example employs one or moremachine (e.g. computer)-readable (hereinafter “computer-readable”)signal-bearing media. The computer-readable signal-bearing media storesoftware, firmware and/or assembly language for performing one or moreportions of one or more embodiments of the invention. Examples of acomputer-readable signal-bearing medium for the financial activitysystem 100 comprise a storage component such as the one or more storagedevices 20. The computer-readable signal-bearing medium for thefinancial activity system 100 in one example can comprise one or more ofa magnetic, electrical, optical, biological, and atomic data storagemedium. For example, the computer-readable signal-bearing medium cancomprise floppy disks, memory devices, magnetic tapes, CD-ROMs,DVD-ROMs, hard disk drives, and electronic memory. In another example,the computer-readable signal-bearing medium comprises a modulatedcarrier signal transmitted over a network comprising or coupled with thefinancial activity system 100, for instance, one or more of a telephonenetwork, a local area network (“LAN”), a wide area network (“WAN”), theInternet, and a wireless network.

VII. Data Structures

In a general sense, data needed for a financial activity may be locatedentirely on site, as contemplated herein. Alternatively, the presentinvention also contemplates situations where administrators of afinancial activity may decide to divide responsibilities with an outsideservice. For example, a credit or other financial service may be engagedto provide credit checks, authorize financially responsible individualsto participate, or even handle all financial matters, including thecollection, payout and other handling of funds. As another alternativecontemplated by the present invention, financial activities may beconducted with authorized brokers, financial institutions as well asregulated exchanges such as stock and futures exchanges. Such activitiesmay reflect predefined financial arrangements. In any of these orpossibly, other instances, the data handling, participant authorizationand other interactions concerning the participant may reside off-site,that is, remote from the operation (and possibly the responsibility) ofthe financial activity.

With reference to FIGS. 2-4, the databases 32 in one example comprise aparticipant database, an administrator or system database, a creditprovider's database, a storm watch database, a rules database and a pushdatabase. The credit provider database contains a list of creditproviders and their accepted methods of payment, as well as any creditrelated information of any type, such as authorization codes usuallyprovided to merchants or the like to authorize transactions acceptableto a credit provider. The credit provider database may also containother financial information associated with the credit provider, such asthe credit provider's identification number and account information. Inthe preferred embodiment, the system administrator in one aspectprovides services similar to that of a merchant selling goods and/orservices to participants. If desired, the system administrator cancomprise a reseller of goods and services such as proprietary weatherreports and cartographic or weather information, as well as maps, formsand other materials relating thereto. In another example, operators ofthe financial activity may rely entirely on outside services to providethe needed credit and other financial arrangements.

The participant database maintains a list of participants and theirassociated personal financial information. The participant databasestores a set of personal payment methods which are registered by theparticipant with a transaction processing service, which in thepreferred embodiment is engaged by the system administrator as anaccommodation to the participant. The participant database furtherincludes information regarding the eligibility of participants toparticipate in the financial activity. In the preferred embodiment, thesystem administrator employs a known screening service to enforce thoserules set down pertaining to restrictions on participation. For example,the system administrator may choose to implement requests by governmentofficials to curtail or otherwise limit transactions originating in orcommunicated to those areas subject to an evacuation order or one ormore legal restrictions. Further details concerning an exemplaryscreening service is provided in U.S. Pat. No. 6,508,710, issued Jan.21, 2003, the disclosure of which is hereby incorporated by reference inits entirety.

For example, a participant could specify by checking a box on a list asto whether he is registering as an individual, corporation, partnership,trust, etc. Depending upon which box is checked, another display willask what the entity's net worth is. Should that net worth meet or exceeda set amount for that particular entity, then the participant will beregistered for that activity based upon a minimum net worth set by thefinancial activity provider or regulated exchanges (i.e.—eligiblecontract participant, institutional trader, retail trader, etc.).

The storm watch database tracks storm activity of interest toparticipants. Included, for example, are circulating storm systems whichhave not yet matured into hurricanes, but which have the potential fordoing so. If desired, historical data concerning previous storm systemsmay be made available to participants, either on an unrestricted basisor at additional cost to the participant.

The administrator database contains data and other information needed tooperate the financial activity. Included, for example, are ongoing“real-time” or “moving” totals of the number of participants, the totalinvested, the number of other participants that have positions (e.g.financial investment units) corresponding to a participant's predictionchoice, and the amount invested by the other participants. If desired,the administrator database can also include real-time estimates ofpayout amounts corresponding to the participant's prediction choice,assuming that the choice proves to be accurate. Such payouts may includeconsideration of some form of minimum return the participant may expect,for those financial activities providing such expectations. Theadministrator database can also include a list of known users who are tobe barred from participating or otherwise restricted in theirparticipation activity. This information can be contained in a separatedatabase, if desired. Also, the administrator database preferablycontains participation statistics and financial statistics, useful inproviding an updated estimate of the cost of doing business foroperating the financial activity. If desired, adjustments to coverfluctuations in overhead costs can be made with regard to futureparticipants.

The rules database contains rules or other principles of operation forthe financial activity. The rules database contains a set of “rules” orprinciples which govern the ongoing financial activity, in a specific orin a general way (e.g. rules defining the authorities, or externalobjective independent information sources to be relied upon for a final,factual decision or conclusion). Examples of such authorities includeexpert governmental agencies responsible for monitoring natural perilevents, as well as weather stations which provide reports. The rulesalso include eligibility requirements, personal financial paymentrequirements, and sliding scales affecting payouts such as timing anddeadlines.

The rules may be wholly or partially public (i.e. available toparticipants) or private (i.e. available only to those authorized by thesystem administrator). In one instance, the rules database also governsthe course of conduct of specific aspects of the financial activity. Forexample, in one instance, the rules include definitions relating to thenatural peril events to be considered by the financial activity, theexternal objective independent information source which managesinformation and determinations concerning a natural peril event whichwill be relied upon during the course of conducting the financialactivity, parameters associated with the natural peril events,especially those parameters which are used to uniquely define eachparticular natural peril event as well as parameters for determiningremuneration points or other value.

If desired, the points or other value pertaining to the participant'sremuneration can be “hidden” or incorporated within a calculation, andneed not be expressed in an explicit reference. In another instance, therules database contains definitions of those participants eligible toengage in the financial activity, as well as those participants whichqualify as finalists (“winners”) eligible or who otherwise qualify forremuneration. In a further instance, the rules database containsprinciples of operation governing transfers between the financialactivity and qualifying participants. The rules database may also governaccess that a participant has to certain information concerning thefinancial activity, such as the number of individuals participating, theaverage or largest financial investments currently being made, and theraw total currently collected for the event of interest.

In another instance, the rules database can include principles ofoperation relating to safety and public interest considerations. Forexample, the rules database can provide for automatic suspension ofoperation upon public announcement of an evacuation order orrecommendations to prepare to evacuate a particular area. The rulesdatabase can provide for selective activity based upon the location ofthe participants. For example, suspension of financial activity can belimited only to those counties or other areas where government safetywarnings have been issued, while allowing financial activity to continuefor those areas not affected by the government warnings.

The push database contains information useful for generating interestand encouraging participant activity. For example, push data can includerecent designations of officially recognized storm systems that maybecome candidates for future investment opportunities. Push data canalso include brief analyses and/or statistics of ongoing or recentnatural peril events. Different amounts of push data and different listsof push participants can be set up by a computer program according topre-defined “trigger levels” such as storm location, intensity andspeed, for example. The push database can also include rules ofoperation pertaining to push data, such as local times during which pushdata is or is not sent.

The data files comprise data information which, preferably, isrelatively static, such as the official designations of natural perilevents to be issued in an upcoming activity season, official andunofficial historical reporting of natural peril events activity andstatistics compiled from historical information, for example. Thishistorical data can be combined with climatological and otherprobabilities to determine investment price and/or payouts. If desired,the data files can be replaced by one of the available databases, or aspecial database can be provided, if desired.

A land area database can contain geographic items such as maps and otherdata relevant to conducting a financial activity. For example, in theUnited States, maps can be provided for those states at risk to ahurricane strike. Preferably, the maps would be “clickable” to allow aparticipant to readily indicate the state of interest. In response, moredetailed maps such as maps of the counties within the state would bedisplayed to the participant and again, would be clickable to provideready indication of the participant's choices of predicted strike areas.If desired, this same functionality can be provided in table form orsome other form convenient for user participation. In addition, a crossreference “finder” tool can be provided to receive Zip code informationor the like, and return with a colored or other visually distinctivearea on the displayed map, or a textual response to the inquiry, readyfor the participants' selection to the indicated. As mentioned, it isgenerally preferred that the maps, tables, or other geographic locationinformation contain a visual indication of those areas which lie outsideof the financial activity, providing a ready indication of ineligibilityto participants surveying their possible choices for a prediction entry.If desired, the geographic location data can be linked to meteorologicalor climatological data for the given area.

A weather database is preferably provided for information concerningweather, meteorological or climatological or other natural forces suchas precursors to earthquakes or volcanic eruptions. The weather databasepreferably contains historical information helpful to those preparing aprediction of future natural peril events.

VIII. Participant Terminals

Turning now to FIG. 3, terminal apparatus 58 comprises one or more ofthe participant terminals 14-18 and includes a communications port 60,one or more processors preferably comprising a central processing unit62 and a memory storage unit 64. Also included is an interface component68 which preferably comprises a display 70 and a data input device 72.Interface component 68 allows a participant or other user 76 tocommunicate with apparatus 58 and in turn with apparatus 13 of thecentral managing system 12. The present invention also recognizes othertypes of devices, such as pda's laptop computers and cellularcommunication devices, as means for conducting financial activitiesaccording to the present invention. As will be seen herein the presentinvention also contemplates an organization of equipment and servicesthat may be loosely be referred to as a network in which third partyservices such as clearing houses or exchanges participate in thefinancial activity, along with the aforementioned participants and theoriginators or managers of the financial activity.

Turning now to FIG. 4, terminal apparatus 80, comprising another exampleof the participant terminals, includes a communication port 60, one ormore processors preferably comprising a central processing unit 62, amemory storage unit 64 and an interface component 84. In the arrangementillustrated in FIG. 4, interface component 84 includes, in addition to adisplay 86 and a data input device 88, a card read/write device 92 andan output device 94 for dispensing a printed receipt, confirming aparticipant's transaction.

With additional reference to FIGS. 5-6, two additional examples ofparticipant terminals are shown. With reference to FIG. 5, participantterminal 200 is shown comprising a display 202 for presentinginformation about the selected natural peril events, a user interfaceintegrated with the display for viewing event information and placinginvestments on a selected natural peril event, an optional cardread/write device 206 for receiving an electronic or magnetic-stripecard encoded with a participant's account information, an optionalticket dispensing device 210 for providing a ticket comprising purchaseinformation for a selected natural peril event and a housing 214 forretaining the display, the user interface, the card read/write deviceand the ticket dispensing device.

The participant terminal 200 also includes a processor and may alsoinclude a speaker (not shown) for playing audio associated with thefinancial activity information. The display preferably comprises a CRTor a flat screen display 218 for displaying information regarding thenatural peril events and preferably, the display comprises atouch-sensitive display, including a touch-sensitive membrane (notshown) in communication with the processor for selecting the desiredinvestment information such as the desired investment in terms ofdollars or the number of financial investment units, as well as“scrolling” between next and previous information. As will be apparentto those skilled in the art, any appropriate type of display may beused.

Turning now to FIG. 6, another embodiment of the at least oneparticipant terminal, generally indicated at 230, is shown comprising adisplay 232 for presenting information about the selected natural perilevent, a user interface 236 for viewing event information and makinginvestments, an optional card read/write device 240 for receiving anelectronic or magnetic-stripe card encoded with a user's accountinformation, an optional ticket dispensing device 242 for providing aticket comprising investment information for a selected natural perilevent and a stand-up type housing 250 for retaining the display, theuser interface, the card read/write device and the ticket dispensingdevice. The participant terminal also includes a processor (not shown)for facilitating financial activity. The participant terminal 230 mayalso include a speaker (not shown) for playing audio associated with thefinancial activity information. The examples shown in FIGS. 5 and 6 areonly exemplary implementations for the at least one participantterminal, and other configurations are also contemplated. For example,the user interface may include a plurality of hardware or softwarebuttons, each identifying different functions for facilitating variousaspects of the financial activity.

IX. Central Managing Apparatus

The central managing apparatus 13 and the participant terminal apparatus(together, referred to as “the apparatus”) in one example comprise aplurality of components such as one or more of electronic components,hardware components, and computer software components. A number of suchcomponents can be combined or divided in the apparatus. An exemplarycomponent of the apparatus employs and/or comprises a set and/or seriesof computer instructions written in or implemented with any of a numberof programming languages, as will be appreciated by those skilled in theart. The apparatus in one example comprises any (e.g., horizontal,oblique, or vertical) orientation, with the description and figuresherein illustrating one exemplary orientation of the apparatus, forexplanatory purposes.

The apparatus, in one example, employs one or more computer-readablesignal-bearing media. The computer-readable signal-bearing media storesoftware, firmware and/or assembly language for performing one or moreportions of one or more embodiments of the invention. Examples of acomputer-readable signal-bearing medium for the apparatus comprise thestorage components 20, 64. The computer-readable signal-bearing mediumfor the apparatus in one example comprises one or more of a magnetic,electrical, optical, biological, and atomic data storage medium. Forexample, the computer-readable signal-bearing medium may comprise floppydisks, magnetic tapes, CD-ROMs, DVD-ROMs, hard disk drives, andelectronic memory. In another example, the computer-readablesignal-bearing medium may comprise a modulated carrier signaltransmitted over a network comprising or coupled with the apparatus, forinstance, one or more of a telephone network, a local area network(“LAN”), a wide area network (“WAN”), the Internet, or a wirelessnetwork.

The present invention also contemplates arrangements where some or allof the central managing services are performed by one or more thirdparties, such as clearing houses as well as professional third partyservices authorized to conduct financial or management functions.

X. Graphical User Interface

Turning now to FIGS. 7 and 8 a-8 f, and initially to FIG. 7, program 30includes one or more subroutines for communicating with a participantlocated at a remote participant terminal. In one instance, program 30includes one or more subroutines for generating one or more screensperforming a number of functions, including sending information to aparticipant, and receiving information from the participant. In FIG. 7,window or screen 300 schematically represents a summary screen forparticipant John Doe, as indicated at 310. As mentioned, screen 300 is asummary screen, and works with a number of supporting screens whichquery the participant for specific information such as the participant'sname, and receives responsive information which is then reviewed forform and content, recorded in one or more databases such as theparticipant database, and is reported in the area 310.

Other supporting screens receive other participant applicationinformation, such as the participant's location of residence or locationof other property holdings, along with information regarding theparticipant's credit information, or approval from an external service,such as a participating brokerage service. Upon approval, eitherinternally or through an exchange clearing organization or the likeservice, the participant's credit and other qualifications, an accountis opened for the participant and details concerning the account, creditqualifications and other related financial information are stored in oneor more databases, such as the participant database. Alternatively,operators of the financial activity may rely entirely on outsideservices to provide the needed credit and other financial arrangements.The steps referred to herein regarding credit check and the like may bereplaced by an authorization from the external service. In otherinstances, operators of the financial activity may interact directlywith an approved exchange service, with financial arrangements beingmade according to rules governing operation of the exchange.

In any event, the summary screen 300 is then presented to theparticipant, confirming the participant's active status in the financialactivity. During this process, one or more queries, multiple-datachoices, multiple activity choices or other interactions with theparticipant are listed in the area 314. If desired, each choicepresented to the participant can have an adjoining checkbox 318,provided for the ready data input into program 30. If desired, one ormore command buttons 320-330 can be provided for the user, to executeone or more commands or otherwise control some portion of data is storedin one or more databases, or to control some portion of program 30allocated to the participant by the system administrator. If desired,area 334 can be provided to display context-sensitive rules of play tothe participant, or to provide appropriate prompts or other helpfulinformation. If desired, checkboxes 336 can be provided adjacent eachentry in window or pane 334 to allow the participant to obtain furtherinformation related to the topic of interest.

In the area 340, the participant is alerted to the current operationbeing performed by program 30. If desired, a sequence of operationsappearing in area 340, along with appropriate responsive indicationsfrom the participant may be listed in area 344. If desired, informationin area 344 can be saved or printed using command buttons 328, 330, thusaffording the participant the opportunity to obtain a written record ofthe activities in either electronic or printed paper form.

Turning now to FIGS. 8 a-8 f, a events of exemplary data input screensare shown in schematic form. These screens pertain generally to theselection of the locations chosen by the participant for investment. Forexample, if the natural peril event is a hurricane, the location may bethe participant's prediction of where a hurricane strike will maketerrestrial contact. Alternatively, the location may be the epicenter ofthe storm's strike, the point of landfall, or a point along the overland track of the tropical cyclone. Landfall can be defined in anynumber of ways. For example, landfall can be measured using the centerof the eye of hurricane, or the eye wall of the hurricane or differentportions of the structure of the hurricane. Referring to FIG. 8 a, ascreen 400 presents a map 402 of a land area, which is preferablysubdivided into smaller portions, each of which may either be selectableby the participant or shaded or otherwise made visually distinctive tothe participant so as to indicate an area which is not eligible for thefinancial activity.

If desired, the rules stored in one or more databases may providefurther information regarding this topic of ongoing activity.Preferably, each subdivided portion of map 402 is selectable by touchscreen, click and point, or by an input pen device, for example. It isalso preferable, in one instance, that the area 406 selected by theparticipant is shaded, colored, or otherwise made visually distinctiveso as to indicate graphically the choice made by the participant. InFIG. 8 a, area 406 is chosen by the participant and receives adistinctive contrasting color value to provide visual feedback to theparticipant. As indicated in FIG. 8 a, the screen 400 also includes aquery to the participant to confirm and finalize the choice of location.

Turning now to FIG. 8 b, screen 410 is presented as a prompt to theparticipant to expand the indicated area so as to include one or moresurrounding areas. In screen 410, an enlarged area 412 surrounding theinitial chosen location 406 (the “collar countries”) is made to flash orblink on and off or undergo a color change. An optional text message ispresented to draw the participant's attention to the advantages ofenlarging the selected location in which a natural peril event strike ispredicted to occur. The participant can indicate additional locations byshift-clicking, for example.

In one instance, it may be desirable to establish a rule of playallowing the use of so-called secondary parameters. These secondaryparameters require a participant to select not only a location of strikeby a natural peril event, but also to indicate some characterizingfactor associated with the natural peril event, such as binary eventsincluding the strength of the tropical storm measured according to anumerical category value, according to the Saffir-Simpson scale, forexample. Another example of a secondary parameter for a tropical stormcould be the strength of the storm, and such is contemplated in FIGS. 8c-8 f. Referring to FIG. 8 c, screen 420 provides notice to theparticipant that a secondary parameter is to be provided, in addition tothe strike location. In FIG. 8 d, a pull-down window 424 is provided inscreen 426 to indicate a range of values to be chosen by the participantas the predicted category strength of the hurricane strike. In FIG. 8 e,it is assumed that a participant has previously enlarged the area ofstrike location to be covered by the chosen investment or “stake”. Inscreen 430, the participant's choice of category 2 is confirmed alongwith an invitation to spread the participant's stake in categorystrength, as well as in land area. In FIG. 8 f, a screen 440 shows theparticipant's selected range of category strength. Following, is ascreen (not shown) which summarizes the participant's stake. Forexample, for the investment indicated in FIG. 8 f, a user has selectednine geographic areas and three category strengths, for a total purchasecost of 27 financial investment units (9 areas×3 strength values). Inone instance, it is generally preferred that this summary total offinancial investment units purchased is reported in area 344 of screen300, shown in FIG. 7.

XI. Methods and Operations

An illustrative description of exemplary operation of the system isconsidered with initial reference to FIGS. 9-12. As will be seen, thefinancial activity discussed here is modeled after a game of skill orthe like. Other types of financial activities will require differentmethods, apparatus and operations.

FIGS. 9-12 indicate a series of steps to be carried out during thecourse of the financial activity. These method steps may be implementedin a number of different ways, including, for example, but notlimitation, execution of program 30 by the central managing apparatus13, and one or more participant terminals. The program 30 may beimplemented by either a general-purpose computer or a special purposeelectronic device, for example. The method steps may be incorporatedinto an article of manufacture such as a data storage device. As will beseen herein, the steps indicated in FIGS. 9-12 indicate that portion offinancial activity as taken from the viewpoint of the systemadministrator.

Referring initially to step 500, the financial activity is initiated byvirtually any appropriate means. For example, if the rules of operationprovide that the financial activity begins at a given date and time, thestart step 500 may be implemented in software that monitors the systemclock and executes program code which publishes invitations toparticipants to engage in financial activity as of the referenced dateand time. Alternatively, start step 500 may be initiated by the systemadministrator pressing a key switch or otherwise activating a switch toinitiate transmission to participants indicating that the financialactivity season has been opened. An event season may be related to asingle natural peril event or a number of different natural peril eventsor a number of portions of an ongoing natural peril event. In oneinstance, an event season is defined by calendar dates, by a number ofoccurrences of a defined natural peril event, or by a mixture of both,or may rely upon a report or other dissemination of data from anexternal objective independent information source.

In step 502 in addition to announcing the opening of the financialactivity season, an optional offer is made to make available certainrules of operation which govern financial activity for the season ofoperation. In step 504 application information is obtained from theparticipants. This information can include, for example, an indicationof the identity of the participant, the participant's residentiallocation or location of property interests, and the participant's creditinformation needed to allow the system administrator to authorizeopening of an account for the participant. Alternatively, theadministrator may rely upon external credit or other financial servicesto take the necessary action culminating in authorization of aparticipant's account. Preferably, the system administrator predefinesacceptance criteria in the rules which govern the financial activity.These rules may include intervention by an external agency, such as anexternal credit agency from which credit is purchased by theparticipant, using the system administrator as a broker, as is currentlydone by many merchants offering goods or services for sale.

In step 506, the application information is reviewed and the decisionmade in step 508 is to either accept or reject the participant'sapplication. For example, a participant's application may be rejectedbecause the participant has failed to disclose a property interestneeded to base financial activity on property losses caused byoccurrence of natural peril events. If the participant's application isrejected, control is passed to step 511 which sends a terminationmessage to the participant and returns control to step 504.

If the participant's application is accepted, control is passed to step510 in which credit information is obtained from the participant. Theparticipant's credit qualifications are reviewed in step 512 and adecision is rendered in step 514 to accept or reject the participantbased upon the added requirements of appropriate credit qualifications.Again, if the participant fails to meet sufficient creditqualifications, control is passed to step 511 which sends a terminationmessage to the participant and then transfers control to step 504. Ifthe decision in step 514 is positive, indicating acceptance of theparticipant's application and credit qualifications, control is passedto step 516 which confirms the active status of the participants withrespect to the financial activity. Such confirmation may be indicated,for example, by a report rendered by screen 300 as explained above, withreference to FIG. 7.

Referring to FIG. 10, in step 518 the prediction entry is requested fromthe participant. In the step, the participant provides informationdefining the investment to be made. After confirming the unique identityof the natural peril event, the participant declares the primaryparameter information which, in one instance, comprises the location ofthe land strike predicted for the natural peril event. Thereafter, theparticipant declares any secondary parameters required by the rules ofoperation, such as the severity of the strike, and the strike duration,for example. In step 520, the prediction entry and other investmentinformation is obtained from the participant and stored in one or moredatabases, for future reference. In one instance, the time at which theinvestment information is obtained is noted and stored along with theparticipant's investment data. In one instance, the amounts of payout orremuneration to a successful participant is weighted according to theamount of time between the investment transaction and occurrence of theevent, with greater time durations being weighted more favorably, on thepremise that later investments have the benefit of accumulated knowledgewhich will benefit the ability to predict occurrence of an event.

The investment information is reviewed in step 522 and judgment is madein step 524 as to whether the investment information is acceptable ornot. If the investment information is rejected in step 524, control ispassed to step 526 which sends an error message to the participant,passing control to step 518 to repeat the information gathering process.If desired, step 526 can cause relevant information to be offered ordisplayed to the participant to help raise the participant's level ofskill in making a prediction. If desired, the participant can be askedto answer a number of questions relating to the skills involved inmaking a prediction for the particular natural peril event.

Assuming that the investment information is in the correct form andmeets other automated criteria, control is passed to step 528 in whichthe investment information is recorded along with the current time. Asmentioned, this information, and resulting decisions to authorize aparticipant may be as simple as receiving a favorable indication from anexternal credit entity, clearing house, or the like service. The systemadministrator has the option of determining at what point in the ongoingfinancial activity a participant is deemed to have completed theinvestment process for the purpose of determining the time differencebetween the investment and occurrence of the natural peril event. Forexample, the times noted in either steps 520 or 528 (or some other timeif desired) can be used. As a further alternative, a systemadministrator may wish to defer appointing an investment time to theparticipant until monies for the transaction have been obtained from theparticipant. As a concession to the participant, the systemadministrator may provisionally appoint an investment time at an earlierstep.

Assuming the investment information has been successfully obtained andrecorded, control is passed to step 530 in which necessary monies arecollected from the participant. If desired, the participant's ability topay can be guaranteed before hand to eliminate any time delay at thispoint in the ongoing financial activity. Further, as mentioned above,these and other financial arrangements may be handled by an externalagent or service, or may be otherwise provided for according to variousregulatory bodies. In step 532, the participant's investment isconfirmed by an entry to the summary screen 300, for example. Withreference to FIG. 11, control is then passed to step 534 in whichcontact is made with an external objective independent informationsource that observes the natural peril event and manages informationconcerning the natural peril event and optionally, renders relateddecisions, such as assigning a severity level according to establishedscales of measure.

Generally speaking, it is preferred that the system administrator not berequired to render decisions concerning occurrence of a natural perilevent, such as primary and secondary event parameters. In one instance,a system administrator provides in the rules of operation that afinancial activity will rely upon a designated external objectiveindependent information source for information concerning the occurrenceand characteristics and other parameters of natural peril events uponwhich investments are to be based. In step 534, connection to anexternal objective independent information source may be initiated oralternatively, data from the external objective independent informationsource previously obtained may be accessed for use by the financialactivity. In step 536 updates to ongoing developments received from theexternal objective independent information source may be posted for thebenefit of existing and prospective participants. In one instance,updates are made on an ongoing “live” basis, either with little or notime delay, or at a minimum, at a time prior to closing of a naturalperil event.

In step 538, information from the external objective independentinformation source is queried to determine if the external objectiveindependent information source has established start of a natural perilevent of interest to the financial activity. If an event has not yetstarted, control is passed to step 536. When the external objectiveindependent information source determines that an event has started, aunique identifier for the natural peril event is assigned and recordedto one more databases. In one instance, the unique identifier isthereafter associated with each investment by a participant concerningthe natural peril event. In step 540, the start of the event is reportedto the participants and the time of event starting as “officially”determined under rules of the system administrator is posted orotherwise made available to participants, and is recorded in one or moredatabases for possible future reference by the financial activity. Instep 544, if the event has not yet closed, continuous updates regardingevent progress are obtained and in one instance, are reported orotherwise made available to the participants.

Once an event has closed, control is passed to step 546 to determine iffinancial activity has ceased. In one example, the system administratorprovides the rules of operation defining the starting and ending timesfor a financial activity season. This can be based upon an arbitrarydate and time, or upon occurrence of a particular event, such asoccurrence of the fourth, fifth, or sixth tropical cyclone since theseason opened. In one instance, closing of one season may be followed byan immediate or delayed opening of a subsequent season. For example, asubsequent season can be declared by the system administrator toaccommodate financial investments based upon occurrence of the fourth,fifth or sixth tropical cyclone occurring in a given hurricane season,as defined by the National Hurricane Center.

Referring to FIGS. 11 and 12, if it is determined in step 546 that aseason has ended, the season of financial activity is closed in step 548and accumulated data and other information is reviewed in step 550 todetermine and identify the finalists which have made successfulpredictions concerning the natural peril event, as provided in the rulesof operation for the financial activity. In step 552 a number ofcalculations are made in preparation for making payouts to thesuccessful finalists. In one instance, calculations are made todetermine the total holdings, the remuneration points per finalist, theadministration and operating fees associated with conducting thefinancial activity, and the amounts of remuneration for each finalist.Remuneration points, in one instance, are based solely upon the primaryparameter, which is preferably the strike location of the natural perilevent. In another instance, remuneration points are determined not onlyby the strike location but by other natural peril event parameters,secondary or otherwise.

If desired, an additional outcome, called a “null” event may be definedfor a financial activity. The null event is an investment choice made toparticipants, giving them the opportunity to invest in the possibleoutcome where there is no U.S. hurricane landfall in the currentoperating period, such as any number of different weather events,seasons or years. Participants investing in the null outcome, forexample, may receive a return at the end of hurricane season or otheroperating period.

In one instance, a single primary parameter is defined by the systemadministrator in the system rules of operation. In another instance, oneor more secondary parameters are also defined in the system rules ofoperation. In a further instance, secondary parameters are assigned alesser weighting than the primary parameter. In any event, the primaryand secondary parameters, if any, can have equal or unequal weighting,as may be desired. In one example, relating to hurricane events,location of the hurricane's strike is defined as a primary parameter,with time delay between the investment time and the time of thehurricane strike at the investment location being defined as a secondaryparameter.

In one instance, the time delay secondary parameter is weighted lessthan the primary parameter. In another instance, severity of the naturalperil event at the predicted location declared by a participant isdefined as a tertiary parameter, and the secondary and tertiaryparameters are assigned unequal weighting. Remuneration points may bedetermined according to a mathematical formula, algorithm, market pricesor other operation which does not require human intervention at the timeof execution. Thus, the formula, algorithm or other operation may beincorporated in an analog or digital electronic circuit or a hydrauliccircuit, for example. If desired, and especially with financial activityproviding a hedge against property losses, remuneration points may bedetermined, contingent upon or otherwise based upon confirmation of theparticipant's property interests.

In step 554 payouts are made to the successful participants, orfinalists. In step 556 the financial activity is closed.

The steps or operations described herein are just exemplary. There maybe many variations to these steps or operations without departing fromthe spirit of the invention. For instance, the steps may be performed ina differing order, or steps may be added, deleted, or modified.

In addition to the above, other types of activities are contemplated. Asmentioned above, a participant may elect to contact a systemadministrator or other service provider to engage in a financialactivity. Investments are paid into a system account to purchasefinancial investment units in the financial activity. Assuming aparticipant's activities are successful and perhaps if certainqualifications are met, payouts are made from the system account to theparticipant. Other types of transactions include, for example, financialinvestment units purchased in the course of the financial activity caneither be uniquely assigned to a participating individual, or they canbe made freely transferable. Accordingly, a financial activity may beorganized such that either payouts must be made to the participantmaking the investment or payouts can be made to any individualpossessing sufficient identification, such as an account number andpassword.

In either example, the investment positions are referred to herein asfinancial investment units or stakes, whether in the nature of aderivative holding or not, can be bought and sold between variousparties, either with or without interaction with personnel associatedwith the financial activity. If desired, the aftermarket activity infinancial investment units transfer can be offered by the operator ofthe financial activity as a service to members of the public. In anyevent, the financial activity, from investment to pay out would becarried out by operators of the financial activity. In anotherembodiment, financial activities can also be carried out between two ormore participants, with the operator of the financial activity providinga service that facilitates financial interactions between the partiesinvolved. Alternatively, the administrators of the financial activitymay elect to engage a brokerage institution, exchange, or clearingorganization, for example, to handle financial and related functions.

XII. Variability Factors

Different factors affecting price and/or payout or settlement of thefinancial activity can be employed as an alternative to, or incombination with, market pricing that depends on market activity of thefinancial operation. Different factors can apply at different times. Forexample, early on in a financial activity, prices may be set accordingto current historical event activity. Prices later on can be setaccording to market activity, as by raising the price of a recentinvestment choice, and by lowering all other available choices.

By way of introduction, two considerations are contemplated in oneinstance, one relating to probability (e.g. one or more probabilities)and the other relating to a calendar or timing of events. In oneinstance, a price or cost variation in the purchase of a financialinvestment unit (or “stake”) representing a quantification of aparticipant's financial involvement is provided. Other variabilityfactors include numerical expressions, either continuous ordiscontinuous, of the impact, predicted impact or risk of an impact ordegree of impact of a natural peril event.

In one instance, assuming a point in time before occurrence of a naturalperil event, participants are able to invest in a financial activity atprices which are set by the financial activity provider, and which varydepending on time and/or on one or more probability factors. A firstcomponent of price variability, in one instance, keeps track of thetiming of the investment. It should be borne in mind that investmentscan be made a long time (e.g. months) before a natural peril event, suchas the time a hurricane would be likely to occur. One purpose of thisvariability factor (namely that of timing), is to encourage investmentsto be made earlier, rather than later. This variability factor, ineffect, preferably operates as a price discount factor, although thevariability factor could also be applied to payout distributions.

One or more probability assessments are preferably made at the time of aparticipant making an investment in the financial activity. Oneprobability assessment preferably takes the form of a probabilitycalculation based on current conditions, of the likelihood of a “hit,”“successful outcome” or “qualification” that a participant's predictionwill occur. For situations involving hurricane natural peril events, theprice of a unit available for purchase by a participant at a given timeis calculated based upon a probability that the county (or othergeographical designation for a purchase unit) chosen by the participantwill suffer a hurricane strike. If desired, probabilities can be basedupon storms other than hurricanes and if desired the strength of thestorm or other factor can be employed to alter the purchase price at anygiven time.

Related to probability-type variability factors are scales or indexesupon which a financial activity may be based, in whole or in part, incombination with other types of variability factors and other treatmentsconsidered herein. Scales can be employed to provide a basis forpricing, payout or other settlement of a financial activity. As will bediscussed in greater detail below, scales can be based upon observednatural activity, as well as characterizations, measurements and otherquantifications thereof. Thus, the scales may be used to introduce acertain measure of variability into the financial activity, which can becombined, if desired, with algorithms or other devices, as discussedherein. Scales play a role in financial activities involving derivativesecurities interests, as where indexes are traded bilaterally ormultilaterally, as discussed herein. Scales are usually termed “indexes”in this context.

As a further consideration, financial activities can take into accountone or more conditional probabilities. The following example employsthree stages of probabilities, with reference to one example of anatural peril event dealing with hurricane activity. It has beenobserved that the average number of hurricane landfalls on the U.S. in agiven year is 1.7. This value is sufficient to define a Poissondistribution, according to conventional techniques, for numbers of U.S.landfalls, which yields a probability of 0.817 that at least one USlandfalling hurricane will occur in a given year. Similarly, from thisPoisson distribution, the conditional probability that there will be atleast two US landfalling hurricanes, given that one has alreadyoccurred, is 0.620.

In one instance, financial investment units in later portions of thefinancial activity (held, for example, for subsequent natural perilevents) can be priced more cheaply than earlier events according toconditional probabilities of K strikes, given that K−1 strikes havealready occurred, so that, in addition, prices go up in subsequentevents when an earlier event closes.

As mentioned, the financial activity, cited as an example of variabilityfactors, incorporates three stages of probability assessment, withdifferent probability treatments being given at each stage. Preferably,three probability treatments are applied to investment price, but couldalso be applied to payout distributions if desired. In a first stage ofprobability assessment, no storms or other precursors to hurricanes arein existence, and the first stage probabilities preferably are based onclimatological relative frequencies.

Second stage probabilities are in play where at least one storm or othernatural peril event exists in the field of interest (usually, ageographic area such as the Atlantic basin) but is far enough away fromthe area of interest (e.g., the coastline of the continental UnitedStates and adjacent coastlines) that no forecasts of imminent impact ofthe natural peril event, such as landfall, can be made. Preferably, forhurricane natural peril events, attention is paid at this stage totropical depressions, and tropical cyclones such as hurricanes and totheir location and tracks at sea. Preferably, in addition to thedistance between the tropical depression and the area of interest,attention is paid to the historical tracks or paths of storms inprevious years that subsequently made landfall in the area of interest.

In the third stage, occurrence of a natural peril event such as theimminent impact of the natural peril event (e.g. landfall of ahurricane) is officially recognized and preferably quantified as to itsimmanency. It is generally preferred that the existence of the thirdstage is declared, based upon an indication of an independent objectiveinformation source, such as the National Hurricane Center/TropicalPrediction Center. For example, a provider of a financial activity maylook to the issuance of hurricane watches, and especially hurricanewarnings from the National Hurricane Center (NHC) where a hurricanewatch is issued when it is determined that hurricane conditions maythreaten an area within 24 to 36 hours. At this point, preparations maybe made for an imminent evacuation, if one is ordered. A hurricanewarning is issued when hurricane conditions (i.e. maximum sustained windspeeds of 74 mph or more) are expected in a specified coastal areawithin 24 hours or less. Local government agencies make independentassessments and independently issue evacuation orders for people in theaffected areas. Notification of these types of events to the financialactivity provider can be used to close further participant activity, oralternatively to trigger a shift from stage two probability assessmentsto stage three probability assessments. The NHC forecasts go out 72 or120 hours into the future, depending on the nature of the forecastproduct. It should be noted that in this scenario, the effect of stagethree probability assessments is intentionally weakened by the calendaror the timing variability factor. Since the natural peril event isimminent, price discounts for unit purchases is preferably very low ornil. If desired, payout penalties can be assessed for stage threeinvestments because of their close timing to occurrence of the naturalperil event.

XIII. Considerations Regarding Pricing

In general, the pricing paid by participants of the financial activityis in one instance, based on the concept that the participants are giventhe option of investing more or less, as they may desire. It ispreferred that this be implemented by offering the participants theability to purchase investment positions in discrete quantities,generally referred to herein as “financial investment units.” In someinstances, depending on the financial activity, these financialinvestment units may be identified as shares or options. However, thepresent invention also contemplates financial activities where theamount of investment trading available to each participant is fixed,with the price for the fixed amount of trading being either constant orchanging throughout the financial activity.

In general, pricing may be held at a fixed level, or may vary throughoutone or more portions, as well as the entirety of the financial activity.For example, operators of the financial activity may choose to varyprice according to variability factors, such as those described herein(e.g. according to an algorithm or according to the timing of theinvestment). As a further alternative, pricing may vary in whole or inpart according to market conditions, with pricing reacting to marketactivity. In such instances, the pricing may vary directly, inproportion to, or in some amount, but in the same general trend as,changes in market activity. Further, market activity can becharacterized in a variety of ways, such as direct relation to the totalnumber of units in play or in some proportional or nonproportionalmanner based on some aspect of market activity. Pricing and/or marketactivity may take into account “current” and “recent” market changes.

Examples of pricing methods are given herein, with reference toexemplary types of financial activity. It will be appreciated, however,that pricing methods can be readily adapted to other types of financialactivities, as well. A detailed consideration of one type of pricingstrategy will now be discussed.

XIV. Pari-mutuel Market with Endogenous Prices

1. Introduction

Pricing operations are discussed for a pari-mutuel market relating tofirst hurricane landfalls, an example of a preferred First LandfallMarket according to principles of the present invention. As will be seenherein, in the most preferred First Landfall Market, an adaptive controlalgorithm is employed for setting prices. The First Landfall Market ispreferably implemented using graphical user interface features of FIG.24 and following (and less preferably, using graphical user interfacefeatures of FIGS. 13-18) as discussed herein. Other natural peril eventscould be chosen, as well. Included are a series of binary options for aset of mutually exclusive and collectively exhaustive events relating tothe location of the next U.S. landfalling hurricane at one of 83 coastalsegments (most are individual counties) spanning the U.S. east and Gulfcoasts from the Mexican to Canadian borders. In the event that nofurther U.S. hurricane landfalls occur in a given hurricane season, an84th event, termed “Null,” is deemed to occur. However, the marketstructure is more general and could be used to support hedging andspeculation in other contexts also.

This First Landfall Market considered here allows participants to hedgeor speculate on the first county where the next hurricane makes landfallin the U.S. by trading the options on an exchange, which will be adesignated contract market under the Commodity Exchange Act. Theseinstruments are commodity options—the commodity being defined inexchange rules to be where a hurricane will make landfall first. Underexchange rules, a market participant selects one of the 84 outcomeswhich the market participant fears (or believes, or both) will be theU.S. county where a hurricane will first make landfall. That marketparticipant is “long” the county selected and “short” all the othercounties. The market participant pays a premium reflecting this combined“call” on the county selected and “put” on all the other counties. Themarket participant can lose only the amount of the paid premium. If thehurricane makes landfall first in the county selected, the optionholders for that county receive a pro-rata share of the combinedproceeds from premia received and deposited with the exchange in apari-mutuel, or mutual risk pool, for all purchases for all counties inthat option series. In other words, purchases of options in all 84outcomes fund the payout to the holders of options for the county wherethe hurricane first makes landfall.

Subsequent to sales of “primary” options, as just described, aconventional bilateral bid/ask market in the options can also besupported. Both primary sales and this secondary market can operatesimultaneously, even though the two will be linked to a degree.

2. Mathematical Exposition of the Market Structure

Mathematically, denote the dollar total in the pari-mutuel, or mutualrisk pool at a time t as M_(t), and denote the number of options thathave been purchased for county k at time t as N^(k) _(t). When it hasbeen determined which of the k=1, . . . , 84 outcomes has occurred, thepayout for each option held in county k is

$\begin{matrix}\begin{matrix}{{W_{t}^{k} = {M_{t}/N_{t}^{k}}},} & {{if}\mspace{14mu}{the}\mspace{14mu}{storm}\mspace{14mu}{has}\mspace{14mu}{first}\mspace{14mu}{landfall}\mspace{14mu}{at}\mspace{14mu}{county}\mspace{14mu} k} \\{{= 0},} & {{otherwise}.}\end{matrix} & (13)\end{matrix}$If t=τ, a time at which the landfall outcome (if any) is known, Equation13 specifies the actual payout per option. At previous times, t<τ,Equation 13 specifies the “indicative” payouts; that is, it indicatesthe payout that would be received if no further purchases were to bemade in any of the outcomes, and outcome k were ultimately to occur.

A pari-mutuel market for hurricane landfalls in a given year begins inJanuary, and may extend through the end of hurricane season, on 30November. Because available information about the eventual location ofthe first hurricane landfall will change substantially during thisperiod, it is not appropriate for the option prices to remain static.Rather, the prices are updated dynamically as such information changes,and in particular are determined in proportion to the (time-evolving)probabilities for each of the outcomes. A straightforward and naturalmeasure for the probability of outcome k at any given time, as assessedin aggregate by the market, is the ratio of the premium risked (theprice) to the potential reward (the indicative payout) per option,

$\begin{matrix}{{v_{t}^{k} = {\frac{p_{t}^{k}}{W_{t}^{k}} = \frac{p_{t}^{k}N_{t}^{k}}{M_{t}}}},} & (14)\end{matrix}$where p_(t) ^(k) is the price paid for an option in outcome k at time t.

Unfortunately, it is not feasible to use the market-assessedprobabilities in Equation 14 for setting prices because of a circularityin definition: Equation 14 specifies market probability as a function ofprice, yet price is determined in proportion to probability. Thisdifficulty is circumvented through the introduction of a set of pricingprobabilities π_(t) ^(k), for the outcomes k at time t. These pricingprobabilities are continually updated in a way that makes them “shadow”the market probabilities in Equation 14. Using these pricingprobabilities π_(t) ^(k), prices are determined according to

$\begin{matrix}\begin{matrix}{{p_{t}^{k} = {\pi_{t}^{k}c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {\pi_{t}^{k} > \beta} \\{{= {\beta\; c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {{\pi_{t}^{k} \leq \beta},}\end{matrix} & (15)\end{matrix}$where c is a constant dollar amount (perhaps c=$1000) called the “par”value; r is an annual interest rate reflecting time value of money,which is introduced in order not to penalize early investors; and jindicates the day of the year (e.g., j=1 for January 1, j=32 forFebruary 1, etc.). Here β is a minimum pricing probability, taken to beβ=0.0001 in the simulations described below. The scaling constant c iscalled “par” because, if the pari-mutuel market is functioning smoothly,an investor purchasing an option for p_(t) ^(k) dollars can expect apayout in the neighborhood of c dollars if county k receives the firstlandfall.

Optionally, the payout given in Equation 13 can be modified to include a“floor”, or guaranteed minimum payout to holders of options in theoutcome that eventually occurs. In this case, Equation 13 is modified toyield

$\begin{matrix}\begin{matrix}{{W_{t}^{k} = {\max\left( {{Fc},{M_{t}/N_{t}^{k}}} \right)}},} & {{if}\mspace{14mu}{the}\mspace{14mu}{storm}\mspace{14mu}{has}\mspace{14mu}{first}\mspace{14mu}{landfall}\mspace{14mu}{at}\mspace{14mu}{county}\mspace{14mu} k} \\{{= 0},} & {{otherwise},}\end{matrix} & (16)\end{matrix}$where the floor F is a guaranteed fraction of the par value, c. In thiscase, the prices in Equation 15 must be modified in order to be able tohonor the floor guarantees, i.e.,

$\begin{matrix}\begin{matrix}{{p_{t}^{k} = {Fc}},} & {{M_{t}/N_{t}^{k}} \leq {Fc}} & \\{{= {\pi_{t}^{k}c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {{{M_{t}/N_{t}^{k}} > {Fc}},} & {\pi_{t}^{k} > \beta} \\{{= {\beta\; c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {{{M_{t}/N_{t}^{k}} > {Fc}},} & {\pi_{t}^{k} \leq {\beta.}}\end{matrix} & (17)\end{matrix}$Of course Equations 16 and 17 reduce to Equations 14 and 15,respectively, when there are no guarantees (F=0).

Before the market is opened, it must be “seeded” with a modest stake ineach of the outcomes. This can be done on the basis of prior (in thecase of the hurricane market, long-term climatological) probabilities π₀^(k). An initial total pool M₀ is apportioned among the 84 outcomesconsistent with Equations 14 and 15, so that N₀ ^(k)=M₀/c, equally foreach of the k outcomes. The result is congruence between the initialmarket and pricing probabilities, v₀ ^(k)=π₀ ^(k).

The pricing probabilities π_(t) ^(k), are updated each time a newpurchase is made. The time index t in this updating process is notchronological time, but rather is incremented with each individualpurchase, and so is equal at any moment to the total number of optionsthat have been purchased in all counties:

$\begin{matrix}{t = {\sum\limits_{k}{N_{t}^{k}.}}} & (18)\end{matrix}$XV. Adaptive Control Algorithm

Following each purchase of an individual option, the pricingprobabilities for all of the outcomes are updated using an adaptivecontrol algorithm:

$\begin{matrix}\begin{matrix}{{\pi_{t}^{i} = {\pi_{t - 1}^{i} + {\alpha_{t}^{k}{\pi_{t - 1}^{k}\left( {1 - \pi_{t - 1}^{i}} \right)}}}},} & {i = k} \\{{= {\pi_{t - 1}^{i}\left( {1 - {\alpha_{t}^{k}\pi_{t - 1}^{k}}} \right)}},} & {i \neq {k.}}\end{matrix} & (19)\end{matrix}$Here the updated pricing probability π_(t) ^(i) at step t for outcome idepends on the pricing probability π_(t−1) ^(k) pertaining to the optionin the outcome (k) that was most recently purchased (at the previoustime, t−1). Accordingly, the first line of Equation 19 is used to updatethe pricing probability for the outcome k most recently purchased, andthe second line is used to update pricing probabilities for all otheroutcomes. Here α_(t) ^(k) is a small adjustment parameter, 0≦α_(t)^(k)<<1, that varies according to the state of the market, as describedbelow. The effect of this updating procedure is that, for α_(t) ^(k)>0,the pricing probability for the outcome in which the last purchase wasmade increases, and the pricing probabilities for the remaining outcomesdecrease. The structure of Equation 19 ensures that the updatedprobabilities are coherent, i.e., 0<π_(t) ^(i)<1 for all outcomes i, andΣ_(i)π_(t) ^(i)=1.

For each new purchase, the adjustment parameter α_(t) ^(k) is chosenaccording to the relationship between the pricing probability π_(t−1)^(k) and the market probability v_(t−1) ^(k) (Equation 14) for theoutcome k just purchased. In particular, α_(t) ^(k) is chosen in orderto move these two probabilities toward equality, allowing the pricingprobabilities π to track, or “shadow” the market probabilities v. Threecases can be distinguished:

-   -   Case I: π_(t−1) ^(k)>v_(t−1) ^(k). Here the pricing probability        for outcome k is too high relative to the aggregate market        opinion from Equation 14. This condition can also be diagnosed        from the relationship between the indicative payout and the par        value (adjusted for time value of money), since

$\begin{matrix}{{{\pi_{t - 1}^{k} > v_{t - 1}^{k}} = \frac{\pi_{t - 1}^{k}c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}{W_{t - 1}^{k}}},{{{so}\mspace{14mu}{that}W_{t - 1}^{k}} > {c\;{{\exp\left\lbrack {{rj}/365} \right\rbrack}.}}}} & (20)\end{matrix}$That is, options for outcome k are overpriced if the indicative payoutis greater than the (interest-adjusted) par value. For this case,increasing the pricing probability for outcome k would produce evengreater separation between the pricing and market probabilities, soα_(t) ^(k)=0 is used in Equation 19, and pricing probabilities for allof the outcomes are unchanged. Any further purchases in outcome k willreduce the indicative payout W^(k) through increases in N^(k), so thatv^(k) will rise toward π^(k). Subsequent purchases in any outcome otherthan k will drive π^(k) downward, toward v^(k).

Further consideration regarding the adaptive control algorithm is givenlater, herein, and is employed in the most preferred manner of settingprices for the First Landfall Market that is preferably implementedusing the graphical user interface of FIGS. 24-49 (and less preferably,the graphical user interface of FIGS. 13-18) as discussed herein.

-   -   Case II: Market equilibrium where π_(t−1) ^(k)=v_(t−1) ^(k).        Here the market is in equilibrium before the purchase of the        next option t for outcome k. This purchase will raise the market        probability for this outcome, v_(t) ^(k)>v_(t−1) ^(k), so the        pricing probability π_(t) ^(k) should increase correspondingly.        Explicit indication of outcome k using superscripts will be        suppressed for notational simplicity). We wish to increase the        pricing probability π_(t), using Equation 19, to match the        increase in v_(t) resulting from the payout dilution for this        outcome produced by the purchase of one additional option.        Therefore,

$\begin{matrix}\begin{matrix}{\pi_{t} = {\pi_{t - 1} + {\alpha_{t}{\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)}}}} \\{= v_{t}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}{W_{t}}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}\left( {N_{t - 1} + 1} \right)}{M_{t - 1} + {\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}\left( {N_{t - 1} + 1} \right)}{{N_{t - 1}c\;{\exp\left( {{rj}/365} \right)}} + {\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}}} \\{= {\frac{\pi_{t - 1}\left( {N_{t - 1} + 1} \right)}{N_{t - 1} + \pi_{t - 1}}.}}\end{matrix} & (21)\end{matrix}$Here use has been made of the fact that, because of the equilibrium atstep t−1, M_(t−1)=N_(t−1) c exp(rj/365). Solving for the equilibriumadjustment parameter,

$\begin{matrix}\begin{matrix}{\alpha_{t} = {\left\lbrack {\frac{\pi_{t - 1}\left( {N_{t - 1} + 1} \right)}{N_{t - 1} + \pi_{t - 1}} - \pi_{t - 1}} \right\rbrack/\left\lbrack {\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)} \right\rbrack}} \\{= \frac{\pi_{t - 1}\left\lbrack {\left( {N_{t - 1} + 1} \right) - N_{t - 1} - \pi_{t - 1}} \right\rbrack}{{\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)}\left( {N_{t - 1} + \pi_{t - 1}} \right)}} \\{= \frac{1}{\left( {N_{t - 1} + \pi_{t - 1}} \right)}} \\{\approx {\frac{1}{N_{t - 1}}.}}\end{matrix} & (22)\end{matrix}$Since, in all realistic cases N_(t)>>π_(t), the appropriate value forthe adjustment parameter is α_(t) ^(k)=1/N_(t−1) ^(k). Thus, anadditional purchase moves prices relatively little if there are alreadya large number of options in existence for outcome k, but results inlarger price changes if there are relatively few such options. Followingthe algebra in Equation 20, this condition can also be diagnosed fromequality of the indicative payout and the (adjusted) par value.

-   -   Case III: π_(t−1) ^(k)<v_(t−1) ^(k). Here the options for        outcome k are under priced, and α_(t) ^(k) should be chosen to        adjust π_(t) ^(k) upward. Intuitively, this adjustment should be        relatively modest (α_(t) ^(k)≈1/N_(t−1) ^(k)) for π_(t−1)        ^(k)≈v_(t−1) ^(k), and increase as the discrepancy between        π_(t−1) ^(k) and v_(t−1) ^(k) increases, until a maximum value        α_(max) is chosen corresponding to the maximum discrepancy        between π_(t−1) ^(k) and v_(t−1) ^(k). This maximum discrepancy        occurs for W_(t−1) ^(k)=Fc. However, the specific form that the        function for α should take between these two endpoints is not        clear. Referring now to FIG. 43. two candidate pricing curves        for specifying the parameter α, in the range Fc≦W≦c are shown.        FIG. 43 shows two candidate pricing curve functions, the        consequences of which are explored through simulation in the        next section. A value for α_(max) must also be chosen. When        N_(t−1) is small, so that α_(max)<1/N_(t−1), then        α_(t)=1/N_(t−1).

In one example of financial activity according to principles of thepresent invention, equation 19 implements an adaptive control mechanism.Adaptive control here concerns the situation when the exact model of thecontrolled system is not known and has to be ‘learnt on line’ whilecontrolling it. Further information regarding adaptive controlmechanisms is given in: 1) K J Åström, B Wittenmark, Adaptive Control,Addison-Wesley, 1995 2) Narendra, K. S., and Balakrishnan, J. “Adaptivecontrol using multiple models,” IEEE Transaction on Automatic Control,”February 1997, Volume: 42, Issue: 2 pp. 171-187 and 3) Bellman, R., andR. Kalaba, “On Adaptive Control,” IRE Transactions on Automatic Control,November 1959, Volume: 4, Issue: 2, pp. 1-9.

Using first landfall of hurricanes as one example, it is preferred touse an adaptive algorithm that “learns” as it works, and repricesoptions in multiple regions according to probabilities that the market“believes” to reflect the landfall risks, which are themselvesinfluenced by meteorological forecasts for hurricane activity. Theadaptive control mechanism is preferred in a one-sided market, ratherthan a traditional bilateral market where there are both buyers andsellers. When changes in forecast information become available to theparticipants in this financial activity (such as updates providing forchanges in a hurricane's potential landfall area, direction of movement,wind speed, strength, etc.), the pricing algorithm is able to reactquickly to the resulting changes in investment positions as selected bythe participants. For example, as a hurricane becomes closer tolandfall, the algorithm reacts to relatively higher buying levels forthe now-more-likely outcomes, by raising prices to appropriate newlevels for those outcomes. Otherwise, earlier investors in thoseoutcomes will be disadvantaged as later buyers with better or morerecent information, to extract value from those early investors bybuying more cheaply into a given location than is justified. This typeof monitor is not preferred in bilateral markets because, wheninformation changes such that an asset has more value than before,potential sellers will hold out for higher (new) prices rather thansettling for the lower (old) price.

Optionally, in another variation, the adjustment parameter alpha inEquation 19 could be defined as a decreasing function of the number ofoptions, Nk, that have been previously purchased in county k. In oneimplementation, since Equation 22 shows that alpha should decrease ininverse proportion to Nk, alpha would be defined as the quotient of afixed constant divided by Nk. This fixed constant is chosen to be largerthan 1 (the value implied by Equation 22), in order for the resultingprices to be able to respond quickly to deviations from marketequilibrium, such as might be brought on by changes in meteorologicalcircumstances that would make some counties more likely, and somecounties less likely, targets than previously.

3. Simulation Example

This section describes stochastic simulations of the pari-mutuelhurricane market, using the 2004 hurricane season through the firstlandfall of Hurricane Charley, as an example. That is, the 2004hurricane season is simulated many times, using different random butconceptually reasonable sequences of investments in the variouscounties. Charley formed in the eastern Caribbean, and tracked south ofJamaica and over western Cuba before making landfall on the west coastof Florida, at Lee county, on 13 August.

The overall flow of money into the pari-mutuel, or mutual risk pool istaken as the fixed but plausible sequence shown in Table 4 (see FIG.46), with figures in millions. Here it is assumed that the initialseeding is $2M, averaging about $25K per each of the 84 outcomes Table 4specifies strong investment interest from January through mid-February,with a relative lull until May, and then an increase again near thebeginning of hurricane season on 1 June. During the hurricane season,investment interest increases beginning on 31 July. The tropicaldepression that becomes hurricane Alex materializes on 31 July, but doesnot make landfall. The tropical depression that will become tropicalstorm Bonnie first appears on 3 August. The tropical depression thatwill become hurricane Charley first appears on 9 August. The dollartotal in the pool when closed by the approach of Hurricane Charley on 12August is $2 B, of which about ⅓ has been invested before the beginningof hurricane season on 1 June, and ⅔ during hurricane season.

This assumed daily sequence of dollar flows is a very challenging onefor the algorithm, on two counts. First, the initial seeding is verylight relative to the funds coming into the pool during the first twoweeks, so that substantial price volatility is expected. Second, 45% ofthe eventual $2 B pours into the fund during the last two days inresponse to the imminent landfall of Hurricane Charley. These newpurchases are concentrated in counties on the west coast of Florida, sothe internal market probabilities v must respond very quickly to theevent if the eventual payout W_(τ) at Lee county is to be maintainednear the par value.

The simulation time step is once daily, meaning that during hurricaneseason only one of the 6-hourly NHC advisories is used to forecast themeteorological risks of first landfall. From 31 July through 2 Augustthese are for Alex, from 3 August through 8 August these are for Bonnie,and for 9 August through 12 August these are for Charley. For eachsimulated 2004 season, the dollars specified for each day in Table 4 areallocated to the 84 outcomes according to a combination ofmeteorological risks and random factors. Specifically, let D(j) be thedollars invested over the entire pool on day j, from the middle columnin Table 4. Define g_(k)(j) to be the random relative allocation of D(j)to county k on day j, so that the money invested in county k on day j is

$\begin{matrix}{{m_{k}(j)} = {\frac{g_{k}(j)}{\sum\limits_{i = 1}^{84}{g_{i}(j)}}{{D(j)}.}}} & (23)\end{matrix}$The random relative allocations g_(k)(j) are gamma-distributed randomvariables, with meanμ_(k)(j)=ω_(k)(j),  (24)where ω_(k)(j) is the forecast probability for county k on day j. Thegamma distributions from which the relative dollar allocations g_(k)(j)are drawn have common coefficient of variation (i.e., standard deviationdivided by mean) CV=1/2, which is independent of both time and county.The result is that simulated investments in counties exhibiting stronger(mean) buying interest on a given day will be more variable from run torun of the simulation. This effect is especially strong during the lastfew days of the simulation, in which the ω_(k)(j) are relatively largefor the counties on the west coast of Florida.

Having defined the dollar allocations on each day, the numbers ofoptions bought for each of the 84 outcomes are determined using themathematics herein. For the specific results reported here, fixed valuesare taken for the parameters F=0.5, r=0.05, and β=0.0001. The effects ofeach of the pricing curves labeled “full logistic” and “half logistic”in FIG. 43 are investigated, using values of α_(max) ranging from0.00001 through 0.005. In addition, three levels of overall buyingvolume are simulated through variation in the par value. For c=$1000relatively large (τ≈1.8×10⁸) numbers of options are purchased overall ina given simulated year. An order of magnitude fewer (τ≈1.8×10⁷) optionsare purchased for c=$10000; and still fewer (τ≈1.8×10⁶) are needed whenand c=$10000 but all the dollar amounts in Table 4 reduced by a factorof 10. For each combination of parameter values, 100 simulated years arecalculated.

In order to induce disequilibrium in the simulated market, options arebought in lots, rather than individually, even though prices arerecalculated after each individual purchase, as per Equation 19. Afterrandom allocation of the day's investment dollars to each county usingEquations 21 and 22, the number of options that could be purchased foreach county are calculated, using prices for the end of the previousday. The lot size for each county for the upcoming day is then 1/50 ofthe median of these numbers of options. Having chosen this lot size forthe day, the simulation program randomly chooses among the counties forwhich the day's dollars have not yet been exhausted, and buys one lot.The result is that counties that have a relatively small random dollarallocation for the day are finished early, so that toward the end of asimulated day the buys are concentrated in the few counties withrelatively large random allocations. This procedure simulates the effectof a few large players investing large sums into those counties with thelarger random allocations for the day.

Referring now to FIG. 44, an example time series of prices for Leecounty, and the adjacent but smaller Charlotte county, are shown for onemodel realization of the pari-mutuel market. Horizontal dash-dot linesthrough 30 July indicate the product of the respective forecastprobabilities ω and the par value of c=$10,000 (τ≈1.8×10⁷), toward whichthe market prices should move as they recover from random perturbationsduring this time period. January volatility results from the initialseeding being small relative to the large sequence of early investments.Large price increases in mid-August reflect the large sums beinginvested in counties on the west coast of Florida as Charley approaches.Prices for the final two days are offscale, and are not plotted forclarity in showing the rest of the time series. The half-logisticpricing curve with α_(max)=0.0005 has been used.

FIG. 44 shows time series of prices for one of the 100 realizationsproduced with c=$10,000 (τ≈1.8×10⁷), using the half-logistic pricingcurve with α_(max)=0.0005. Prices are shown for Lee county (red), andthe adjacent but smaller Charlotte County (black). The dash-dot linesindicate the levels, given by the product ωc, toward which the pricesshould move as they recover from random perturbations, during the period1 January through 30 July. Appreciable price volatility is evident inJanuary as the market responds to the very large sums, relative to thesmall initial seeding, that are invested during that time. Subsequently,until the meteorological probabilities ω change on 31 July, theseinternally generated market prices correctly track the levels that theyshould move toward (given the dollar allocations specified by Equations21 and 22), confirming the stability of the internal pricing mechanismdescribed above.

FIG. 45 shows average payouts and price volatility for Lee county, using(a) the full logistic, and (b) the half logistic pricing curves fromFIG. 43; as functions of the adjustment parameter α_(max). Volatility iscalculated as the standard deviation of end-of-day prices between 11January and 30 July, averaged over the 100 realizations for eachparameter combination. Parameter combinations for which the averagepayouts (solid lines) are near the par value reflect good functioning ofthe market. Values of α_(max) that are too small result in averagepayouts that are unacceptably small, as a result of prices not adjustingsufficiently quickly to the large flux of money into the market duringthe last two simulated days. Reduced average payouts for large values ofα_(max) reflect price increases that are too strong, with the resultthat prices for other counties decrease too much, allowing payoutdilution. Not surprisingly, volatility (dashed lines) increasesmonotonically with increases in α_(max). Therefore, the optimal α_(max)appears to be a compromise between prices responding quickly enough tomaintain payouts near par, versus responding slowly enough to suppressexcessive price volatility.

Referring now to FIG. 45, average payouts (solid) and volatility(dashed) for Lee County, are shown using (a) the full logistic, and (b)the half-logistic pricing curves shown in FIG. 42, as functions of theadjustment parameter α_(max). All quantities are expressed as a fractionof the par value, c.

The simulated market responses to the two pricing curves from FIG. 43are similar, but overall results for the half-logistic pricing curve(FIG. 45 b) exhibit lesser sensitivity to choice of α_(max), and lowerprice volatility overall. Both panels in FIG. 45 exhibit a dependence ofmarket response on the overall volume, τ, suggesting that a betterpricing curve than either of the two shown in FIG. 43 could probably befound. However, these results indicate that choosing the half-logisticpricing curve together with α_(max)=0.0005 yields a market that behaveswell over a wide range of possible conditions.

In these simulations, threatened breaches of the payout floor Fc,requiring implementation of the first line of Equation 17 for pricing,occurred only in the first 2 weeks of the simulated years, or during thelast few days. For the most part, parameter combinations yieldingaverage payouts within 80% to 85% of par exhibited few or no suchthreatened breaches in August. Between these two difficult times, thesimulated market was well able to adjust internally to the day-to-dayvariations in the amounts invested and their relative random allocationsamong the counties.

The price volatility and threatened floor breaches occurring very earlyin the simulations are nearly unavoidable given the relatively quitesmall initial seeding of the market. Clearly these would be smaller ifearly investment interest is not as large as assumed here, and of coursecould be reduced by seeding the pool with a larger initial sum. However,a less costly and likely more profitable approach would be for theseeding agency to monitor early market performance closely, and act tosuppress the price volatility that leads to threatened floor breaches bybuying the most under priced counties. Stability could be ensured inthis way through purchase of relatively few of the 84 outcomes, and atprices that would be very favorable relative to their eventual expectedpayouts.

XVI. Application of Pari-Mutuel Market with Endogenous Prices to FirstHurricane Landfall Markets

16.1. Introduction

Operation of a pari-mutuel First Hurricane Landfall Market involvinganother series of binary options for a set of mutually exclusive andcollectively exhaustive events is discussed. This operation is similarin some ways to the operations set out above, but presents the operatorof the financial activity with a number of different options. Theseevents also relate to the location of the next U.S. landfallinghurricane at one of 83 coastal segments (most are individual counties)spanning the U.S. east and Gulf coasts from the Mexican to Canadianborders. In the event that no further U.S. hurricane landfalls occur ina given hurricane season, an 84th event, termed “Null,” is deemed tooccur.

The market structure herein is also more general and could be used tosupport hedging and speculation in other contexts. For example, thismarket structure allows participants to hedge or speculate on the firstcounty where the next hurricane makes landfall in the U.S. by tradingthe options on an exchange, which will be a designated contract marketunder the Commodity Exchange Act. These instruments are commodityoptions—the commodity being defined in exchange rules to be where ahurricane will make landfall first.

Under exchange rules, a market participant selects one of the 84outcomes which the market participant fears (or believes, or both) willbe the U.S. county where a hurricane will first make landfall. Thatmarket participant is “long” the county selected and “short” all theother counties. The market participant pays a premium reflecting thiscombined “call” on the county selected and “put” on all the othercounties. The market participant can lose only the amount of the paidpremium. If the hurricane makes landfall first in the county selected,the option holders for that county receive a pro-rata share of thecombined proceeds from premia received and deposited with the exchangein a pari-mutuel, or mutual risk pool, for all purchases for allcounties in that option series. In other words, purchases of options inall 84 outcomes fund the payouts to the holders of options for thecounty where the hurricane first makes landfall.

Additionally, a “floor” on the payout to option holders in the affectedcounty can be supported, which is expected to be especially appealing toretail investors who would enter the market to hedge against actualproperty- and other storm-related losses. Under this mechanism, aparticipant's hedge against hurricane landfall in a particular countycan be guaranteed a minimum monetary return, conditional on landfall inthat county, which amount would be specified at the time of theinvestment. The pricing algorithm is structured such that thepari-mutuel payouts, including these conditional guarantees, areentirely self-funded, so that the exchange assumes no risk.

Subsequent to sales of “primary” options, as just described, aconventional bilateral bid/ask market in the options can also besupported. Both primary sales and this secondary market can operatesimultaneously, even though the two will be linked to a degree.

16.2. Mathematical Exposition of the Market Structure

The dollar total in the pari-mutuel, or mutual risk pool at a time t isdenoted as M_(t), and the number of options that have been purchased forcounty k at time t are denoted as N^(k) _(t). Upon been determination ofwhich of the 84 outcomes has occurred, the payout for each option heldfor that outcome is

$\begin{matrix}\begin{matrix}{{W_{t}^{k} = {M_{t}/N_{t}^{k}}},} & {{if}\mspace{14mu}{the}\mspace{14mu}{storm}\mspace{14mu}{has}\mspace{14mu}{first}\mspace{14mu}{landfall}\mspace{14mu}{at}\mspace{14mu}{county}\mspace{14mu} k} \\{{= 0},} & {{otherwise}.}\end{matrix} & (25)\end{matrix}$If t=τ, a time at which the landfall outcome (if any) is known, Equation25 specifies the actual payout per option. At previous times, t<τ,Equation 25 specifies the “indicative” payouts; that is, it indicatesthe payout that would be received if no further purchases were to bemade in any of the outcomes, and outcome k were ultimately to occur.

A pari-mutuel market for hurricane landfalls in a given year begins inJanuary, and may extend through the end of hurricane season, on 30November. Because available information about the eventual location ofthe first hurricane landfall will change substantially during thisperiod, it is not appropriate for the option prices to remain static.Rather, the prices are updated dynamically as such information changes,and in particular as market activity adjusts to the changinginformation, as reflected by a set of time-evolving “pricingprobabilities” π_(t) ^(k) for the outcomes k at time t. FIG. 41,discussed below, shows an example of a display of such information to aninvestor, along with other data. These pricing probabilities arecontinually updated in a way that makes them converge toward, or“shadow,” the aggregate market opinion of the outcome probabilities,according to an algorithm that will be described shortly. Using thesepricing probabilities π_(t) ^(k), prices are determined according to

$\begin{matrix}\begin{matrix}{{p_{t}^{k} = {\pi_{t}^{k}c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {\pi_{t}^{k} > \beta} \\{{= {\beta\; c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {{\pi_{t}^{k} \leq \beta},}\end{matrix} & (26)\end{matrix}$where c is a constant dollar amount (perhaps c=$1000) called the “par”value; r is an annual interest rate reflecting time value of money,which is introduced in order not to penalize early investors; and jindicates the day of the year (e.g., j=1 for January 1, j=32 forFebruary 1, etc.). Here β is a minimum pricing probability, taken to beβ=0.0001 in the simulations described below. The scaling constant c iscalled “par” because, if the pari-mutuel market is functioning smoothly,an investor purchasing an option for p_(t) ^(k) dollars can expect apayout in the neighborhood of c dollars if county k receives the firstlandfall.

Optionally, the payout given in Equation 25 can be modified to include a“floor”, or guaranteed minimum payout to holders of options in theoutcome that eventually occurs. In this case, Equation 25 is modified toyield

$\begin{matrix}\begin{matrix}{{W_{t}^{k} = {\max\left( {{Fc},{M_{t}/N_{t}^{k}}} \right)}},} & {{if}\mspace{14mu}{the}\mspace{14mu}{storm}\mspace{14mu}{has}\mspace{14mu}{first}\mspace{14mu}{landfall}\mspace{14mu}{at}\mspace{14mu}{county}\mspace{14mu} k} \\{{= 0},} & {{otherwise},}\end{matrix} & (27)\end{matrix}$where the floor F is a guaranteed fraction of the par value, c. In thiscase, the prices in Equation 26 must be modified in order to be able tohonor the floor guarantees, i.e.,

$\begin{matrix}\begin{matrix}{{p_{t}^{k} = {\pi_{t}^{k}c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {{{M_{t}/N_{t}^{k}} > {Fc}},} & {\pi_{t}^{k} > \beta} \\{{= {\beta\; c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}},} & {{{M_{t}/N_{t}^{k}} > {Fc}},} & {\pi_{t}^{k} \leq \beta} \\{{= {Fc}},} & {{M_{t}/N_{t}^{k}} \leq {{Fc}.}} & \end{matrix} & (28)\end{matrix}$Of course Equations 27 and 28 reduce to Equations 25 and 26,respectively, when there are no guarantees (F=0).

Before the market is opened, it must be “seeded” with a modest stake ineach of the outcomes. This can be done on the basis of prior (in thecase of the hurricane market, long-term climatological) probabilities π₀^(k). An initial total pool M₀ is apportioned among the 84 outcomesconsistent with the initial pricing probabilities quantifying the risk,given by the ratio of price to indicative payout,

$\begin{matrix}{\pi_{0}^{k} = {\frac{p_{0}^{k}}{W_{0}^{k}} = {\frac{\pi_{0}^{k}{cN}_{0}^{k}}{M_{0}}.}}} & (29)\end{matrix}$Thus, N₀ ^(k)=M₀/c, equally for each of the k outcomes.

The pricing probabilities π_(t) ^(k), are updated each time a newpurchase is made. The time index t in this updating process is notchronological time, but rather is incremented with each individualpurchase, and so is equal at any moment to the total number of optionsthat have been purchased in all counties:

$\begin{matrix}{t = {\sum\limits_{k}{N_{t}^{k}.}}} & (30)\end{matrix}$Following each purchase of an individual option, the pricingprobabilities for all of the outcomes are updated using a variant of theRobbins-Monro stochastic approximation algorithm:

$\begin{matrix}\begin{matrix}{{\pi_{t}^{i} = {\pi_{t - 1}^{i} + {\alpha_{t}^{k}{\pi_{t - 1}^{k}\left( {1 - \pi_{t - 1}^{i}} \right)}}}},} & {i = k} \\{{= {\pi_{t - 1}^{i}\left( {1 - {\alpha_{t}^{k}\pi_{t - 1}^{k}}} \right)}},} & {i \neq {k.}}\end{matrix} & (31)\end{matrix}$This is an adaptive control mechanism, in which the aggregate of marketopinion regarding (the possibly time-evolving) probabilities for each ofthe possible outcomes is learned in response to the sequence ofpurchases made by market participants. Here the updated pricingprobability π_(t) ^(i) at step t for outcome i depends on the pricingprobability π_(t−1) ^(k) pertaining to the option in the outcome (k)that was most recently purchased (at the previous time, t−1).Accordingly, the first line of Equation 31 is used to update the pricingprobability for the outcome k most recently purchased, and the secondline is used to update pricing probabilities for all other outcomes.Here α_(t) ^(k) is a small adjustment parameter, 0<α_(t) ^(k)<<1, thatvaries according to the state of the market, as described below. Theeffect of this updating procedure is that the pricing probability forthe outcome in which the last purchase was made increases, and thepricing probabilities for the remaining outcomes decrease. The structureof Equation 31 ensures that the updated probabilities are coherent,i.e., 0<π_(t) ^(i)<1 for all outcomes i, and Σ_(i)π_(t) ^(i)=1.

For each new purchase, the adjustment parameter α_(t) ^(k) is inverselyproportional to the number of options previously purchased in thatoutcome,

$\begin{matrix}{\alpha_{t}^{k} = {\frac{H}{N_{t}^{k}}.}} & (32)\end{matrix}$If few options have been bought previously, then an additional purchasewill move prices relatively more than if many options are already inexistence. Sections following show that, if the market is inequilibrium, H=1 in Equation 32. However, in practice one should chooseH>1 in order for prices to be able to respond quickly to deviations frommarket equilibrium, such as might be brought on by changes in externalinformation (e.g., the meteorological situation) relevant to the market.Simulations indicate that 30≦H≦40 produce a smoothly operating marketwith relatively small price volatility.16.3. Simulation Example

This section describes stochastic simulations of the pari-mutuelhurricane market, using the 2004 hurricane season through the firstlandfall of hurricane Charley as an example. That is, the 2004 hurricaneseason is simulated many times, using different random but conceptuallyreasonable sequences of investments in the various counties. Charleyformed in the eastern Caribbean, and tracked south of Jamaica and overwestern Cuba before making landfall on the west coast of Florida, at Leecounty, on 13 August.

The overall flow of money into the pari-mutuel, or mutual risk pool istaken as the fixed but plausible sequence shown in Table 5, with figuresin millions. Here it is assumed that the relatively small initialseeding is $2M, averaging about $25K per each of the 84 outcomes. Table5 specifies strong investment interest from January throughmid-February, with a relative lull until May, and then an increase againnear the beginning of hurricane season on 1 June. During the hurricaneseason, investment interest increases beginning on 31 July. The tropicaldepression that becomes hurricane Alex materializes on 31 July, but doesnot make landfall. The tropical depression that will become tropicalstorm Bonnie first appears on 3 August. The tropical depression thatwill become hurricane Charley first appears on 9 August. The dollartotal in the pool when further investment in this option series isclosed by the approach of Hurricane Charley on 12 August is $2 B, ofwhich about ⅓ has been invested before the beginning of hurricane seasonon 1 June, and ⅔ during hurricane season.

This assumed daily sequence of dollar flows is a very challenging onefor the pricing algorithm, on two counts. First, the initial seeding isvery light relative to the funds coming into the pool during the firsttwo weeks, so that substantial price volatility is expected initially.Second, nearly a quarter of the eventual $2 B pours into the fund duringthe last two days in response to the imminent landfall of hurricaneCharley. These new purchases are concentrated in counties on the westcoast of Florida, so the pricing probabilities must respond very quicklyto the event if the eventual payout W_(τ) at Lee county is to bemaintained near the par value.

The simulation time step is once daily, meaning that during hurricaneseason only one of the 6-hourly NHC advisories is used to forecast themeteorological risks of first landfall. From 31 July through 2 Augustthese are for Alex, from 3 August through 8 August these are for Bonnie,and for 9 August through 12 August these are for Charley. For eachsimulated 2004 season, the dollars specified for each day in Table 5 areallocated to the 84 outcomes according to a combination ofmeteorological risks and random factors. Specifically, let D(j) be thedollars invested over the entire pool on day j, from the middle columnin Table 5. Define g_(k)(j) to be the random relative allocation of D(j)to county k on day j, so that the money invested in county k on day j is

$\begin{matrix}{{m_{k}(j)} = {\frac{g_{k}(j)}{\sum\limits_{i = 1}^{84}{g_{i}(j)}}{{D(j)}.}}} & (33)\end{matrix}$The random relative allocations g_(k)(j) are gamma-distributed randomvariables, with meanμ_(k)(j)=ω_(k)(j),  (34)where ω_(k)(j) is the forecast probability for county k on day j. Thegamma distributions from which the relative dollar allocations g_(k)(j)are drawn have common coefficient of variation (i.e., standard deviationdivided by mean) CV=1/2, which is independent of both time and county.The result is that simulated investments in counties exhibiting stronger(mean) buying interest on a given day will be more variable from run torun of the simulation. This effect is especially strong during the lastfew days of the simulation, in which the ω_(k)(j) are relatively largefor the counties on the west coast of Florida.

Having defined the dollar allocations on each day, the numbers ofoptions bought for each of the 84 outcomes are determined using themathematics herein. For the specific results reported here, fixed valuesare taken for the parameters F=0.5, r=0.05, and β=0.0001. The parameterH, controlling the step size in the adaptive control of prices, isvaried through the range 1-100. In addition, four levels of overallbuying volume are simulated through variation in the par value. Forc=$100 a very large (τ≈1.8×10⁹) number of options is purchased overallin a given simulated year. An order of magnitude fewer (τ≈1.8×10⁸)options are purchased for c=$1000; and still fewer (1.8×10⁷ and 1.8×10⁶)are needed when and c=$10,000 and $100,000, respectively. For eachcombination of parameter values, 100 simulated years are calculated.

Simulated option purchases are made in lots, rather than individually,even though prices are recalculated after each individual purchase, asper Equation 31. After random allocation of the day's investment dollarsto each county using Equations 33 and 34, the number of options thatcould be purchased for each county are calculated, using prices for theend of the previous day. The lot size for each county for the upcomingday is then 1/50 of the median of these numbers of options. Havingchosen this lot size for the day, the simulation program randomlychooses among the counties for which the day's dollars have not yet beenexhausted, and buys one lot. The result is that counties that have arelatively small random dollar allocation for the day are finishedearly, so that toward the end of a simulated day the buys areconcentrated in the few counties with relatively large randomallocations. This procedure simulates the effect of a few large marketparticipants investing large sums into those counties with the largerrandom allocations for the day.

FIG. 44 shows a time series of prices for one of the 100 realizationsproduced with c=$1000 (τ≈1.8×10⁸), using H=30. Prices are shown for Leecounty, and the adjacent but smaller Charlotte County. The dash-dotlines indicate the levels, given by the product ωc, toward which theprices should move as they recover from random perturbations, during theperiod 1 January through 30 July. Appreciable price volatility isevident in January as the market responds to the very large sums,relative to the small initial seeding, that are invested during thattime. Subsequently, until the meteorological probabilities (ω=0.0106 forLee county, and (ω=0.0021 for Charlotte county) change on 31 July, theseinternally generated market prices correctly track the levels that theyshould move toward (given the random dollar allocations specified byEquations 33 and 34), confirming the stability of the internal pricingmechanism described herein. Similarly, on the final two days of thesimulation, the respective meteorological probabilities for Lee andCharlotte counties are approximately 0.11 and 0.04, and the pricesadjust quite rapidly to levels consistent with these values.Importantly, the pricing algorithm is able to recover from the initialvolatility that arises because of the very thin initial seeding, andconverge toward the simulated market consensus “opinion,” even thoughthat is masked by considerable randomness in the allocation ofinvestments among the counties. The initial volatility could be reducedby a larger initial seeding of the market, but this example emphasizesthe robustness of the pricing adaptation algorithm in Equation 31. Thisinitial volatility could also be reduced by introducing a ceiling on theparameter α, although probably at the expense of a more frequent need toinvoke the payout floor protection (third line of Equation 28).

FIG. 44 shows a time series of prices for Lee county, and the adjacentbut smaller Charlotte county, during one model realization of thepari-mutuel market. Horizontal dash-dot lines through 30 July indicatethe product of the respective forecast probabilities ω and the par valueof c=$1000 (τ≈1.8×10⁸), toward which the market prices should move asthey recover from random perturbations during this time period. Januaryvolatility results from the initial seeding being small relative to thelarge sequence of early investments. Large price increases in mid-Augustreflect the large sums being invested in counties on the west coast ofFlorida as Charley approaches. H=30 has been used in Equation 32.

Table 6 shows the effect on market performance of different choices forthe parameter H, in connection with different overall levels of marketvolume (controlled by different choices for the par, c). Tabulated areaverage returns for Lee county, as a fraction of par, and standarddeviation of these returns over 100 simulated years, although similarresults are obtained for the other counties also. Defining theadjustment parameter α in Equation 31 as being inversely proportional toN^(k) results in similar average returns for a given value of H,regardless of the level of overall volume. In contrast, when a constantvalue for α is used (i.e., no dependence on N^(k), results not shown),larger α's are needed when market volume is small, and smaller α's areneeded when market volume is large.

For the larger values of H in Table 6 the standard deviation of payoutcan be rather large. For the smaller values of H the price adjustmentsrespond too slowly to the large influx of investment into the westernFlorida counties in the final days of the simulation, so that thepari-mutuel, or mutual risk pool is diluted by the rush of lateinvestment that is allowed at prices that are too low, and accordinglyaverage payouts are substantially below par. This dilution does notoccur for moderate values of H, indicating that the market mechanism isresistant to manipulation attempts when the pricing adjustment parameteris defined appropriately. The very large investments in the final twodays specified in Table 5 can be interpreted as simulating the actionsof large speculators seeking to profit from overwhelming the market justbefore the hurricane landfall. But for such attempts to be successful itwould be necessary for them to extract value from earlier investors, sothat payouts near par imply failure of this manipulation strategy.

Asterisks in Table 6 indicate parameter combinations for which the 50%payout floor was never challenged in any of the 100 simulated years (forany of the counties, not just Lee), which would have required use of thethird line of Equation 28. “Plus” symbols in Table 6 indicate threecases where the 50% floor was challenged for a single county, during thefirst few days of January. Together, these cases coincide with the rangeof H for which minimum payout variability is also achieved throughoutthe range of overall market volumes.

Finally, the rightmost column in Table 6 shows price volatility for Leecounty, measured as the standard deviation of end-of-day prices for theperiod 11 January (to exclude early, very high price volatility derivingfrom the light initial market seeding) through 30 July (just before thefirst tropical depression is declared, changing the meteorologicalprobabilities), and averaged over the 100 simulated years. These areshown only for c=$100, but results for other cases are comparable. Notsurprisingly, the volatility increases monotonically with H, since thestep size in the price adjustment algorithm (Equation 31) increases withH. The optimal H will be large enough for average payouts to be nearpar, but as small as possible consistent with this condition in order tominimize price volatility. The results in Table 6 suggest that 30≦H≦40is an appropriate range. A typical screen display presenting investmentinformation to an investor is shown in FIG. 41.

16.4 Derivation of the Correct Adjustment Parameter α at EconomicEquilibrium

This section treats the case of a market in equilibrium, in whichEquation 29 holds at time t−1 for all outcomes k. In this section,explicit indication of outcome k using superscripts will be suppressedfor notational simplicity.

Let v_(t)=p_(t)/W_(t) be the outcome probability implied by the ratio ofrisk (price) to potential reward (indicative payout), as in Equation 29.In one example, the objective is to increase the pricing probabilityπ_(t), using Equation 31, to match the increase in v_(t) resulting fromthe payout dilution for this outcome produced by the purchase of oneadditional option. Therefore,

$\begin{matrix}\begin{matrix}{\pi_{t} = {\pi_{t - 1} + {\alpha_{t}{\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)}}}} \\{= \nu_{t}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}{W_{t}}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}\left( {N_{t - 1} + 1} \right)}{M_{t - 1} + {\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}}} \\{= \frac{\pi_{t - 1}c\;\exp\;\left( {{rj}/365} \right)\left( {N_{t - 1} + 1} \right)}{{N_{t - 1}c\;{\exp\left( {{rj}/365} \right)}} + {\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}}} \\{= {\frac{\pi_{t - 1}\left( {N_{t - 1} + 1} \right)}{N_{t - 1} + \pi_{t - 1}}.}}\end{matrix} & (35)\end{matrix}$Here use has been made of the fact that, because of the equilibrium atstep t−1, M_(t−1)=N_(t−1) c exp(rj/365). Solving for the equilibriumadjustment parameter,

$\begin{matrix}\begin{matrix}{\alpha_{t} = {\left\lbrack {\frac{\pi_{t - 1}\left( {N_{t - 1} + 1} \right)}{N_{t - 1} + \pi_{t - 1}} - \pi_{t - 1}} \right\rbrack/\left\lbrack {\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)} \right\rbrack}} \\{= \frac{\pi_{t - 1}\left\lbrack {\left( {N_{t - 1} + 1} \right) - N_{t - 1} - \pi_{t - 1}} \right\rbrack}{{\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)}\left( {N_{t - 1} + \pi_{t - 1}} \right)}} \\{= \frac{1}{\left( {N_{t - 1} + \pi_{t - 1}} \right)}} \\{\approx {\frac{1}{N_{t - 1}}.}}\end{matrix} & (36)\end{matrix}$This final approximation will be a very close one, because in realisticcases N_(t)>>π_(t).

Further discussion of this type of market is discussed below withreference to a First Landfall Market With Price-Setting Adaptive ControlAlgorithm.

XVII. Theoretical Discussion of a Price Setting Algorithm for NonbinaryPrediction Markets

17.1 Introduction

In a prediction market, investors buy contracts on one of severalpossible outcomes. The contracts bought on the winning outcome receive apayoff at the end. The following discussion considers a predictionmarket with an arbitrary finite number of outcomes and presents atheoretical validation of a price setting algorithm that automaticallyupdates the prices for all the outcomes. Three models of investorbehavior are considered in which investors use a set of commonly heldrisk-neutral probabilities to decide which outcome to buy the nextcontract for. In one of the models the payoff is pari-mutuel. For eachof the three cases a theorem is proved that under that model thenormalized price vector converges to the vector of probabilities.

In standard prediction markets, buyers and sellers are paired up totrade binaries, contracts whose payoffs depend on two possible outcomes.The buyer of outcome 1 pays P1 betting that outcome 1 will occur and theseller (of outcome 1, buyer of outcome 2) pays P2 betting that it willnot. Whoever ends up being correct receives P1+P2. Clearly p1/(P1+P2) isthe buyer's (risk-neutral) probability for outcome 1 and P2/(P1+P2) isthe seller's probability for outcome 1.

In one aspect, the present invention is directed to a nonbinaryprediction market that works as follows:

(a) There is an event that will end with one of n≧2 outcomes beingrealized.

(b) Investors can buy one contract at a time for anyone of the noutcomes from a market maker who publicly quotes a price for everyoutcome. Contracts are held to maturity.

(c) At the end, every contract that was bought for the winning outcomewill receive a payoff.

A market-making algorithm is presented herein that automatically updatesthe price of every outcome. This discussion includes a theoreticalanalysis of the algorithm's performance under three models of investorbehavior. These models specify ways in which investors decide whichoutcome to buy next, based on a commonly held set of risk-neutralprobabilities.

Consideration is given from the perspective of a market maker, althoughits relevance will be readily appreciated by other market participants.In the first two models the payoff is fixed, so the market maker assumessome risk. He must rely on his prices to attract buyers to all theoutcomes so that his liability is covered. In the third model the payoffis pari-mutuel: the pool is divided equally among the winning contracts.Here, attracting buyers to more outcomes is the only practical way tomake their potential payoffs larger.

It will be noted that making all outcomes equally attractive isequivalent to making the prices proportional to the (risk-neutral)probabilities. It is proved that the normalized prices from thealgorithm indeed converge to the probabilities, under all three models.Thus the algorithm automatically learns the probabilities from theinvestors' actions. This algorithm seems special in the sense that everyvariation we considered caused it to lose the learning property.

17.2 Notation and Pricing Algorithm

In the following Q will denote the set of all probability vectors x=(x1,. . . , xn) on n outcomes, meaning that xi≧0 for all i and Σ_(i)x^(i)=1,and Q+ will be the set of x

Q such that x_(t) ^(i)>0 for all i.

If the market maker's prices are too high investors won't buy contractsand if they are too low, revenues will be small. The pricing algorithmpresented herein attempts to maximize revenues for the market maker. Oneprinciple of the algorithm's operation is that a vector xt

Q should be kept updated so as to provide, based upon estimates frominvestors' actions, the empirically established probabilities that areguiding their decisions, and that a price vector of the form pt=λxtshould be quoted, where λ>1 is some constant.

Let t be an integer index that starts at t=1 and is incremented by onewhenever a contract is sold.

If 100 contracts for outcome i are bought together, the price is stillincremented after each individual contract, so the total price paid willbe the same as if 100 consecutive but separate contracts on the sameoutcome were bought.

Together with Q and Q+, the following terminology will be used allthroughout:

-   -   t=1.2, . . . time index of when the t-th contract sells,    -   x_(t)=state vector just before contract t sells, an element of        Q,    -   p_(t) ^(i)=λx_(t) ^(i), price for contract t if it is for        outcome i, λ≧1.    -   k(t)=index of outcome for which contract t sells.    -   x_(t)*=x_(t) ^(k(t)), a space-saving notation.    -   v_(t) ^(i)=1 if contract t is bought for outcome i and 0        otherwise,    -   α_(t)=adaptation parameter, a small number such that 0<α_(t)<1.        At the start, the state vector x₁εQ⁺ is set arbitrarily. From        then on, after contract t is sold, x_(t) is revised to        x_(t+1)εQ⁺ using x_(t+1)=(1−β_(t)) x_(t)+/β_(t)v_(t), where        β_(t)=α_(t)x_(t)*. That is,

$x_{t + 1}^{i} = {{{\left( {1 - \beta_{t}} \right)x_{t}^{i}} + {\beta_{t}v_{t}^{i}}} = \left\{ \begin{matrix}{{x_{t}^{i} + {\beta_{t}\left( {1 - x_{t}^{i}} \right)}},} & {{{if}\mspace{14mu}{contract}\mspace{14mu} t\mspace{14mu}{was}\mspace{14mu}{for}\mspace{14mu}{outcome}\mspace{14mu} i},} \\{{\left( {1 - \beta_{t}} \right)x_{t}^{i}},} & {{if}\mspace{14mu}{contract}\mspace{14mu} t\mspace{14mu}{was}\mspace{14mu}{not}\mspace{14mu}{for}\mspace{14mu}{outcome}\mspace{14mu}{i.}}\end{matrix} \right.}$Every time a contract is sold, the algorithm raises one price and lowersthe other ones. The choice β_(t)=α_(t)x_(t)* is fundamental. Forinstance, with a deterministic β_(t) the algorithm does not learncorrectly. This seemed mysterious until it was discovered that whenβ_(t)=α_(t)x_(t)* the algorithm can be expressed

Algorithms of this type are sometimes called stochastic approximationalgorithms. They are of the form xt+1=xt+αtyt, where xt is anapproximation we are trying to improve gradually and yt is some noisyestimate of the direction in which xt should move, for example a noisygradient in an optimization problem. The novel algorithm underconsideration here works with sequences of probability vectors. Theoremsgiven herein are proved from scratch using at most a standard martingaleresult, and for the most part, can not be proved simply by quotingoff-the-shelf results.

One may wonder if investors could select the outcome they buy at each tin such a way that the sequence of transacted prices p_(t) ^(k(t)) goesdown to zero so fast that total revenues remain bounded, Σ_(t≧1)p_(t)^(k(t))<∞. Lemma 2 in the appendix implies that no such buying strategyexists, that this sum is always infinite.

In practice we need λ>1, for otherwise some arbitrage would be possibleand the investor models we will use, which will seem reasonable only ifthere is no arbitrage, would be unrealistic. Equivalently, one couldquote p_(t)=x_(t) and divide the payoffs by λ>1. An example of anarbitrage strategy is to buy contracts 1 through n successively, in thatorder. The cost of doing that would bex _(t) ¹ +x _(t+1) ² + . . . +x _(t+n−1) ^(n) <x _(t) ¹ +x _(t) ² + . .. +x _(t) ^(n)=1.This strategy takes advantage of the fact that after buying contract ithe algorithm reduces the price of every other contract. For thesimplest payoff type where winning tickets receive one dollar, thisstrategy would give the investor a sure payoff of one dollar for a priceof less than one dollar.

This particular arbitrage opportunity can be removed by quotingp_(t)=λx_(t) with λ>1 sufficiently large, for example with λ≧1/(1−α)when we know that α_(t)≦α for all t. Indeed, using the inequality

${{\prod\limits_{s = 1}^{t}\left( {1 - a_{s}} \right)} \geq {1 - {\sum\limits_{s = 1}^{t}a_{s}}}},$which holds when a_(s)≧0 for all s, as discussed in the proof of Lemma 1in the appendix, one gets

$\begin{matrix}{{\lambda{\sum\limits_{k = 1}^{n}x_{t + k - 1}^{k}}} = {\lambda\left\lbrack {x_{t}^{1} + {x_{t}^{2}\left( {1 - {\alpha_{t}x_{t}^{1}}} \right)} + {{x_{t}^{3}\left( {1 - {\alpha_{t}x_{t}^{1}}} \right)}\left( {1 - {\alpha_{t + 1}{x_{t}^{2}\left( {1 - {\alpha_{t}x_{t}^{1}}} \right)}}} \right)} + \ldots}\mspace{11mu} \right\rbrack}} \\{\geq {\lambda{\sum\limits_{i = 1}^{n}{x_{t}^{i}{\prod\limits_{j = 1}^{i - 1}\left( {1 - {\alpha\; x_{t}^{j}}} \right)}}}} \geq {\lambda{\sum\limits_{i = 1}^{n}{x_{t}^{i}\left( {1 - {\alpha{\sum\limits_{j = 1}^{i - 1}x_{t}^{j}}}} \right)}}} \geq {\lambda{\sum\limits_{i = 1}^{n}{x_{t}^{i}\left( {1 - \alpha} \right)}}}} \\{= {{\lambda\left( {1 - \alpha} \right)}.}}\end{matrix}$

Figuring out how large λ needs to be in order to remove all possiblearbitrage opportunities will not be done here, since it will be assumedthat such a value of λ is being used.

Again, without arbitrage opportunities available, investor modelsconsidered herein will seem reasonable. What is fed back to thealgorithm only involves x_(t). For this reason and because investors'choices as determined by anyone of our three models do not depend on λ,as one will be able to verify readily, the rest of this discussion willassume that λ=1 and call x_(t) the price vector.

17.3 Investor Models

Let qεQ⁺ and suppose investors use (risk-neutral) probability q^(i) foroutcome i. They buy contracts one at a time. The market maker does notknow q so he revises his price vector x_(t) after every sale. Thus, ifcontract t is bought for outcome i, the investor will pay x_(t) ^(i) forit. (Recall from Section 2 that the price vector is p_(t)=λx_(t) but weare assuming without loss of generality that λ=1).

Based on q and the payoff type, investors will be modeled homogeneouslyin one of three ways:

-   Investor Model 1: The payoff to every winning contract is one dollar    and each contract t is bought for the outcome i that offers the best    expected payoff per dollar, q^(i)/x_(t) ^(i). If two or more    outcomes have the best expected payoff, any one of them is chosen.-   Investor Model 2: The payoff itself is irrelevant to our analysis.    All we know is that, on average, investment funds are allocated    across outcomes proportionately to their probabilities because    investor interest is proportional to them. That idea will be    implemented in a way that the selection of the outcome for contract    t is independent of the history.-   Investor Model 3: The payoff is of pari-mutuel style, meaning that    the winning contracts will share the total amount of money that was    invested in all the outcomes. Each contract t is bought for the    outcome i that offers the best expected return assuming contract t    will be the last one to be sold (equivalently, that the distribution    of contracts across outcomes will remain static).

Two important types of α_(t) are constant ones and those that decay tozero slowly, in the sense that Σ_(t≧1)α_(t)=∞. If α_(t)=α is constantthen x_(t) only converges to a neighborhood of q whose size iscontrolled by α. If α_(t) decays to zero appropriately slowly thenx_(t)→q with probability one. The first case is important in practicalproblems where q might be changing with time (slowly) and there is aneed to keep α_(t) away from zero so the algorithm does not lose itsability to track q. These two types of at will be dealt with in aunified manner.

17.4 Investor Model 1: Constant Payoffs

In this section it is assumed that the investor who buys contract tchooses an outcome that maximizes the expected payoff q^(i)/x_(t) ^(i).It is shown that x_(t) converges to a small neighborhood of q.

-   THEOREM 1: Let qεQ⁺ and for tε{1, 2, . . . } let x_(t)εQ⁺ and    α_(t)ε(0, 1). Assume that Σ_(t)α_(t)=∞, that contract t is bought    for an outcome k=k(t)ε{1 , . . . , n} such that x_(t)    ^(k)/q^(k)=max_(i)x_(t) ^(i)/q^(i) and that x_(t) is updated to    x_(t+1) using the pricing algorithm, which we now write as    x _(t+1) ^(i) =x _(t) ^(i)(1+α_(t)(v _(t) ^(i) −x _(t) ^(k(t)))),    i=1, . . . , n,  (37)    where v_(t) ^(i)=1 if i=k(t) and v_(t) ^(i)=0 if i≠k(t). For every    ε>0 define    t_(ε)=min {t≧1:α_(u)≦ε for all u≧t},    which could give t_(ε)=∞ unless α_(t)→0. Note that if α_(t)=α is    constant then t_(ε)<∞ only if ε≧α.

(a) For every ε>0 such that t_(ε)<∞ there is a time T such that|x _(t) ^(i) −q ^(i)|≦ε for all i and all t≧T.  (38)

(b) if α_(t)→0 then x_(t)→q.

PROOF: Part (b) is implied by part (a). We will prove (38) byestablishing the stronger inequalities(1+ε)q ^(i) −ε≦x _(t) ^(i)≦(1+ε)q ^(i) for all i and all t≧T.  (39)

Moreover, the left inequality in (39) is a consequence of the right one,since by the latter

$\begin{matrix}{x_{t}^{i} = {{{1 - {\sum\limits_{j \neq i}x_{t}^{j}}} \geq {1 - {\left( {1 + \varepsilon} \right){\sum\limits_{j \neq i}q^{j}}}}} = {{1 - {\left( {1 + \varepsilon} \right)\left( {1 - q^{i}} \right)}} = {{- \varepsilon} + {\left( {1 + \varepsilon} \right){q^{i}.}}}}}} & \;\end{matrix}$

Therefore only the right half of (39) remains to be proved.

Note that

$\begin{matrix}{{\frac{q^{k}}{x_{t}^{k}} = {{\max\limits_{i}\frac{q^{i}}{x_{t}^{i}}} \geq 1}},} & (40)\end{matrix}$since otherwise we would have x_(t) ^(i)>q^(i) for all i and hence thatΣ_(i)x_(t) ^(i)>Σ_(i)q^(i)=1. Therefore k=k(t)≠i whenever q^(i)/x_(t)^(i)<1.

When t is incremented the ratios q^(zi)/x_(t) ^(z) change as follows:

$\begin{matrix}{{\frac{q^{k}}{x_{t + 1}^{k}} = {{\frac{q^{k}}{x_{t}^{k}}\left( \frac{1}{1 + {\alpha_{t}\left( {1 - x_{t}^{k}} \right)}} \right)} > {\frac{q^{k}}{x_{t}^{k}}\left( \frac{1}{1 + \alpha_{t}} \right)}}},} & (41) \\{{\frac{q^{j}}{x_{t + 1}^{j}} = {{\frac{q^{j}}{x_{t}^{j}}\left( \frac{1}{1 - {\alpha_{t}x_{t}^{k}}} \right)} > \frac{q^{j}}{x_{t}^{j}}}},{j \neq {k.}}} & (42)\end{matrix}$Let ε>0 be such that t_(ε)<∞ and take any outcome i. If for some t≧t_(ε)

$\begin{matrix}{{\frac{q^{i}}{x_{t}^{i}} \geq \frac{1}{1 + \varepsilon}},} & (43)\end{matrix}$

then the same is true at t+1, since by (40), (41) and t≧t_(ε)

${\frac{q^{i}}{x_{t + 1}^{i}} > \frac{1}{1 + \alpha_{t}} \geq {\frac{1}{1 + \varepsilon}\mspace{14mu}{if}\mspace{14mu} i}} = k$

and by (42) and (43)

$\frac{q^{i}}{x_{t + 1}^{i}} > \frac{q^{i}}{x_{t}^{i}} \geq {\frac{1}{1 + \varepsilon}\mspace{14mu}{if}\mspace{14mu} i} \neq {k.}$By induction, if (43) holds at some t≧t_(ε), it holds also for allsubsequent t. Therefore, if the second inequality in (39) is false forall T, it means that there is an i such that

$\begin{matrix}{\frac{q^{i}}{x_{t}^{i}} < {\frac{1}{1 + \varepsilon}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} t} \geq {t_{\varepsilon}.}} & (44)\end{matrix}$

But by (40) this would imply that outcome i is never bought once t=t_(ε)and hence that for all t>t_(ε)

$x_{t}^{i} = {{x_{t_{\varepsilon}}^{i}{\prod\limits_{s = t_{\varepsilon}}^{t - 1}\left( {1 - {\alpha_{s}x_{s}^{k{(s)}}}} \right)}} = {{\exp\left( {\sum\limits_{s = t_{\varepsilon}}^{t - 1}{\ln\left( {1 - {\alpha_{s}x_{s}^{k{(s)}}}} \right)}} \right)} \leq {{\exp\left( {- {\sum\limits_{s = t_{\varepsilon}}^{t - 1}{\alpha_{s}x_{s}^{k{(s)}}}}} \right)}.}}}$Since by Lemma 2 the last term converges to zero, (44) would becontradicted. Therefore (39) has to be true for some T.REMARK. Part (a) of Theorem 1 tells us that if it is desirable to havex_(t) ^(i) to converge to within a certain εof q_(t) ^(i) for all i thenunder Investor Model 1 a sequence α₁, α₂, should be chosen thateventually dips below ε and stays there. For example, the constantsequence α_(t)=ε would work, as would the sequence α_(t)=c/t. The latterone would work for every ε>0.17.5 Investor Model 2: Proportional InvestmentsIn this model, aggregate investor interest in each outcome isproportional to its probability. In particular, it is assumed that theinvestor that buys contract t selects outcome i with probability

${r_{t}^{i} = \frac{q^{i}/x_{t}^{i}}{\sum\limits_{j}{q^{j}/x_{t}^{j}}}},$independent of the history, given x_(t). The state vector x_(t) is thusa discrete-time Markov process with state space Q.

To see that this scheme allocates funds proportionately to theprobabilities, note that the allocation to outcome i will be x_(t) ^(i)with probability r_(t) ^(i) and 0 with probability (1−r_(t) ^(i))Therefore, denoting by E_(t)[•] expectation conditional on x_(t) but notyet on the chosen outcome for contract t, the conditional expectedallocation to outcome i resulting from the purchase of contract t is

$\begin{matrix}{{{E_{t}\left\lbrack {{allocation}\mspace{14mu}{to}\mspace{14mu}{outcome}\mspace{14mu} i} \right\rbrack} = {{x_{t}^{i}r_{t}^{i}} = \frac{q^{i}}{\sum\limits_{j}{q^{j}/x^{j}}}}},} & (45)\end{matrix}$which is indeed proportional to q^(i).

-   THEOREM 2: Let qεQ⁺ and x₁εQ⁺ and for each tε{1, 2, . . . } let    α_(t)ε(0.1). Define a Markov chain {x_(t)εQ⁺: t=1, 2, . . . } by at    each t choosing k(t)ε{1, . . . , n} randomly with k(t)=i having    probability r_(t) ^(i) and setting    x _(t+1) ^(i) =x _(t) ^(i)(1+α_(t)(v _(t) ^(i) −x _(t) ^(k(t)))).    i=1, . . . , n,  (46)    where v_(t) ^(i)=1 if i=k(t) and v_(t) ^(i)=0 if i≠k(t).

(a) Let T be a positive integer and define ε₀=min {q¹, . . . , q^(n),x_(T) ¹, . . . , x_(T) ^(n)}>0. Then for every ε<ε₀ and every s, t suchthat T≦s≦t,

${P\left( {{\max\limits_{s \leq u \leq t}{{x_{t} - q}}} > {\varepsilon + {{{x_{s} - q}}{\prod\limits_{u = s}^{t - 1}\left( {1 - {\left( {\varepsilon_{0} - \varepsilon} \right)\alpha_{u}}} \right)}}}} \right)} \leq {\frac{4{\sum\limits_{u = s}^{t - 1}\alpha_{u}^{2}}}{\varepsilon^{2}}.}$

(b) If

${\sum\limits_{t \geq 1}\alpha_{t}} = {{\infty\mspace{14mu}{and}\mspace{14mu}{\sum\limits_{t \geq 1}\alpha_{t}^{2}}} < \infty}$then with probability one x_(t)→q.

-   PROOF: For all t    x _(t+1) =x _(t)+α_(t) z _(t+1), where z_(t+1) ^(z)=x_(t)    ^(zi)(v_(t) ^(z)−x_(t)*), i=1, . . . , n.  (47)    Writing E_(t)[•]=E[•|x_(t)] and calculating as in (45),

$\begin{matrix}{{E_{t}\left\lbrack z_{t + 1}^{i} \right\rbrack} = {{E_{t}\left\lbrack {x_{t}^{i}\left( {v_{t}^{i} - x_{t}^{*}} \right)} \right\rbrack} = {{x_{t}^{i}{E_{t}\left\lbrack {v_{t}^{i} - x_{t}^{*}} \right\rbrack}} = {{x_{t}^{i}\left( {r_{t}^{j} - {\sum\limits_{j}{r_{t}^{j}x_{t}^{j}}}} \right)} = {\frac{q^{i} - x_{t}^{i}}{\sum\limits_{j}{q^{j}/x_{t}^{j}}}.}}}}} & (48)\end{matrix}$Define the martingale {M_(t), t=1, 2, . . . } by

$\begin{matrix}{{M_{1} = {{0\mspace{14mu}{and}\mspace{14mu} M_{t}} = {{\sum\limits_{s = 1}^{t - 1}{\Delta\; M_{s}}} = {\sum\limits_{s = 1}^{t - 1}{\alpha_{s}\left( {z_{s + 1} - {E_{s}\left\lbrack z_{s + 1} \right\rbrack}} \right)}}}}},{t \geq 2.}} & (49)\end{matrix}$Then (47) can be written as

$\begin{matrix}{x_{t + 1} = {x_{t} + {\alpha_{t}{E_{t}\left\lbrack z_{t + 1} \right\rbrack}} + {\alpha_{t}\left( {{\left( {z_{t + 1} - {E_{t}\left\lbrack z_{t + 1} \right\rbrack}} \right) = {{x_{t} + {\alpha_{t}\left( \frac{q^{i} - x_{t}^{i}}{\sum\limits_{j}{q^{j}/x_{t}^{j}}} \right)} + {\Delta\; M_{t}}} = {x_{t} + {\gamma_{t}\left( {q^{i} - x_{t}^{i}} \right)} + {\Delta\; M_{t}}}}},\mspace{20mu}{{{where}\mspace{20mu}\gamma_{t}} = {\frac{\alpha_{t}}{\sum\limits_{j}{q^{j}/x_{t}^{j}}} \in {\left( {0,\alpha_{t}} \right).}}}} \right.}}} & (50)\end{matrix}$Note that Σ_(j)q^(j)/x_(t) ^(j)≧1, since one cannot have q^(j)<x_(t)^(j) for every j because q, x_(t)εQ⁺.With probability one|ΔM _(t) ^(i)|=α_(t) |z _(t+1) −E _(t) [z _(t+1)]|≦α_(t)(|z _(t+1) |+E_(t) [|z _(t+1)|)≦2α_(t),since |z_(t+1)|²=Σ_(i)|z_(t+1) ^(i)|²≦Σ_(i)(x_(t) ^(i))²≦Σ_(i)x_(t)^(i)=1 and hence |z_(t+1)|≦1. Therefore, for any 1≦s≦t and any ε>0, by astandard martingale inequality.

$\begin{matrix}{{{P\left( {{\max\limits_{s \leq u \leq t}{{M_{u} - M_{s}}}} > \varepsilon} \right)} \leq \frac{E\left\lbrack \left( {M_{t} - M_{s}} \right)^{2} \right\rbrack}{\varepsilon^{2}}} = {\frac{\sum\limits_{u = s}^{t - 1}{E\left\lbrack {{\Delta\; M_{u}}}^{2} \right\rbrack}}{\varepsilon^{2}} \leq {\frac{4{\sum\limits_{u = s}^{t - 1}\alpha_{u}^{2}}}{\varepsilon^{2}}.}}} & (51)\end{matrix}$The martingale inequality is a special case of part (i) of theorem 3.8of chapter 1 of Karatzas and Shreve (1991). It can also be found asinequality (1.4) in section 4.1 of Kushner and Yin (2003), although aproof is not given there.

Subtracting q from both sides of (50)x _(t+1) −q=(x _(t) −q)(1−γ_(t))+ΔM _(t),  (52)so by induction

$\begin{matrix}{\begin{matrix}{{x_{t} - q} = {{\left( {x_{s} - q} \right){\sum\limits_{u = s}^{t - 1}\left( {1 - \gamma_{u}} \right)}} + {\sum\limits_{u = s}^{t - 1}{\Delta\; M_{u}{\prod\limits_{v = {u + 1}}^{t - 1}\left( {1 - \gamma_{v}} \right)}}}}} \\{{= {{{\left( {x_{s} - q} \right){\prod\limits_{u = s}^{t - 1}\left( {1 - \gamma_{u}} \right)}} + {\sum\limits_{u = s}^{t - 1}{f_{u + 1}^{t - 1}\Delta\;{M_{u}.\mspace{14mu} 1}}}} \leq s \leq t}},}\end{matrix}{where}{f_{t}^{t - 1} = {{1\mspace{14mu}{and}\mspace{14mu} f_{u + 1}^{t - 1}} = {{{\sum\limits_{v = {u + 1}}^{t - 1}{\left( {1 - \gamma_{v}} \right)\mspace{14mu}{for}\mspace{14mu} u}} + 1} \leq {t - 1.}}}}} & (53)\end{matrix}$

Let ε>0 and 1≦s≦t and suppose that {x_(t), M_(t)} is a sample path of(x, M) such that

$\begin{matrix}{{{M_{u} - M_{s}}} = {{{\sum\limits_{u = s}^{t - 1}{\Delta\; M_{u}}}} \leq {\varepsilon\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} s} \leq u \leq {t.}}} & (54)\end{matrix}$Then, since the f_(u) ^(t−1) are positive and increase as u increases,

$\begin{matrix}\begin{matrix}{{{\sum\limits_{u = s}^{t - 1}{f_{u + 1}^{t - 1}\Delta\; M_{u}}}} = {{{f_{s + 1}^{t - 1}\left( {\sum\limits_{u = s}^{t - 1}{\Delta\; M_{u}}} \right)} + {\sum\limits_{v = {s + 1}}^{t - 1}{\left( {f_{v + 1}^{t - 1} - f_{v}^{t - 1}} \right){\sum\limits_{u = v}^{t - 1}{\Delta\; M_{u}}}}}}}} \\{= {{f_{s + 1}^{t - 1}\left( {M_{t} - M_{s}} \right)} + {\sum\limits_{v = {s + 1}}^{t - 1}{\left( {f_{v + 1}^{t - 1} - f_{v}^{t - 1}} \right)\left( {M_{t} - M_{v}} \right)}}}} \\{{\leq {\varepsilon\;\left( {f_{s + 1}^{t - 1} + {\sum\limits_{v = {s + 1}}^{t - 1}\left( {f_{v + 1}^{t - 1} - f_{v}^{t - 1}} \right)}} \right)}} = {{\varepsilon\; f_{t}^{t - 1}} = {\varepsilon.}}} \\{\leq {{f_{s + 1}^{t - 1}{{M_{t} - M_{s}}}} + {\sum\limits_{v = {s + 1}}^{t - 1}{\left( {f_{v + 1}^{t - 1} - f_{v}^{t - 1}} \right){{M_{t} - M_{v}}}}}}}\end{matrix} & (55)\end{matrix}$In light of (53), this shows that for any path of (x, M) such that (54)holds,

$\begin{matrix}{\mspace{79mu}{{{{x_{u} - q}} \leq {\varepsilon + {{{x_{s} - q}}{\prod\limits_{u = s}^{t - 1}\left( {1 - \gamma_{u}} \right)}}}},\mspace{14mu}{1 \leq s \leq u \leq {t.\mspace{79mu}{Therefore}}}}} & (56) \\{{P\left( {{\max\limits_{s \leq u \leq t}{{x_{u} - q}}} > {\varepsilon + {{{x_{s} - q}}{\prod\limits_{u = s}^{t - 1}\left( {1 - \gamma_{u}} \right)}}}} \right)} \leq {{P\left( {{\max\limits_{s \leq u \leq t}{{M_{u} - M_{s}}}} > \varepsilon} \right)}.\mspace{14mu} 1} \leq s \leq {t.}} & (57)\end{matrix}$Besides (56) one also gets that for all t≧s there are al a_(t) ^(i)ε[−ε,ε] and b_(t)ε(0, 1) such thatx _(t) ^(i) −q ^(i) =a _(t) ^(i) +b _(t)(x _(s) ^(i) −q ^(i)), i−1, . .. , n.Using this with T in place of s givesx _(t) ^(i) ≧q ^(i) +a _(t) ^(i) +b _(t)(x _(T) ^(i) −q ^(i)), i−1, . .. , n.  (58)For every i, either q^(i)≦x_(T) ^(i) or x_(T) ^(i)≦q^(i). In the firstcase (58) gives x_(t) ^(i)≧q^(i)−ε and in the second case x_(t)^(i)≧q^(i)−ε−(q^(i)−x_(T) ^(i))=x_(T) ^(i)−ε. Hence x_(t) ^(i)≧ε₀−ε forall i, where ε₀=min {q^(zi), x_(T) ^(zi): i=1, . . . , n}. Therefore forany 0<ε<ε₀

$\gamma_{t} = {{\frac{\alpha_{t}}{\sum\limits_{j}{q^{j}/x_{t}^{j}}} \geq \frac{\alpha_{t}}{\sum\limits_{j}{q^{j}/\left( {\varepsilon_{0} - \varepsilon} \right)}}} = {\left( {\varepsilon_{0} - \varepsilon} \right){\alpha_{t}.}}}$This, (57) and (51) prove part (a).

To prove part (b) write c=α₀−ε>0 and note that as t→∞

${\prod\limits_{u = s}^{t - 1}\;\left( {1 - {c\;\alpha_{u}}} \right)} = \left. {{\exp\left( {\sum\limits_{u = s}^{t - 1}{\ln\left( {1 - {c\;\alpha_{u}}} \right)}} \right)} \leq {\exp\left( {{- c}{\sum\limits_{u = s}^{t - 1}\alpha_{u}}} \right)}}\rightarrow 0. \right.$Here the inequality In(1+r)≦r for r>−1 and the hypothesis Σ_(t)α_(t)=∞.Letting t→∞ in (46) one obtains

${P\left( {{\sup\limits_{u \geq s}{{x_{u} - q}}} > \varepsilon} \right)} \leq {\frac{\sum\limits_{u = s}^{\infty}\alpha_{u}^{2}}{\varepsilon^{2}}.}$By hypothesis the sum is finite, so letting s→∞ gives

${P\left( {{\lim\limits_{s\rightarrow\infty}{\underset{u \geq s}{\;\sup}{{x_{u} - q}}}} > \varepsilon} \right)} = {{P\left( {\bigcap\limits_{s > 1}\left\{ {{\sup\limits_{u \geq s}{{x_{u} - q}}} > \varepsilon} \right\}} \right)} = {{\lim\limits_{s\rightarrow\infty}{P\left( {{\sup\limits_{u \geq s}{{x_{u} - q}}} > \varepsilon} \right)}} = 0.}}$Finally, since

$\mspace{20mu}{{\left\{ x_{t}\rightarrow q \right\} = {\left\{ {{\lim\limits_{s\rightarrow\infty}{\underset{u \geq s}{\;\sup}{{x_{u} - q}}}} = 0} \right\} = {\bigcap\limits_{\varepsilon > 0}\left\{ {{\lim\limits_{s\rightarrow\infty}{\underset{u \geq s}{\;\sup}{{x_{u} - q}}}} \leq \varepsilon} \right\}}}},\mspace{20mu}{then}}$${{P\left( x_{t}\rightarrow p \right)} = {{P\left( {\bigcap\limits_{\varepsilon > 0}\left\{ {{\lim\limits_{s\rightarrow\infty}{\underset{u \geq s}{\;\sup}{{x_{u} - q}}}} \leq \varepsilon} \right\}} \right)} = {{\lim\limits_{\varepsilon|0}{P\left( {{\lim\limits_{s\rightarrow\infty}{\underset{u \geq s}{\;\sup}{{x_{u} - q}}}} \leq \varepsilon} \right)}} = {{\lim\limits_{\varepsilon|0}1} = 1}}}},$

which is the statement of part (b).

17.6 Investor Model 3: Pari-Mutuel Payoffs

Suppose now that the payoff is pari-mutuel, meaning that the winningcontracts share the total revenues equally. That is, each winningcontract receives M_(T)/N_(T) ^(i) dollars, where i is the winningoutcome, T is the total number of contracts sold and

${M_{t} = {\sum\limits_{s = 1}^{t}x_{s}^{k{(s)}}}},$revenues collected from the first t contracts.N_(t) ^(i)=number of contracts between 1 and t that sold for outcome i.It is assumed now that the investor that buys contract t chooses it foran outcome k that offers the best expected payoff assuming that no morecontracts will be sold after t. That is, k is such that

$\begin{matrix}{{\left( \frac{M_{t - 1} + x_{t}^{k}}{N_{t - 1}^{k} + 1} \right)\frac{q^{k}}{x_{t}^{k}}} \geq {\left( \frac{M_{t - 1} + x_{t}^{i}}{N_{t - 1}^{i} + 1} \right)\frac{q^{i}}{x_{t}^{i}}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{i.}}} & (59)\end{matrix}$Up to a rounding error (that is small for large t), this is equivalentto assuming that the distribution of contracts across outcomes willremain static.

The following theorem is analogous to Theorem 1 but refers to the caseof pari-mutuel payoffs.

Also, α is constant here. We don't yet have an interesting result fortime-dependent α_(t). Note that the inequalities of part (b) are of thegeneral formφ₁(α)q ^(i)−ψ₁(α)≦x _(t) ^(i)≦φ₂(α)q ^(i)+ψ₂(α),whereφ₁(α)↑1, ψ₁(α)↓0, φ₂(α)↓1 and ψ₂(α)↓0 as α↓0,and that the inequalities in part (a) of Theorem 1 were of this form too(with

instead of α).

-   THEOREM 3: Let q    Q⁺, α    (0,1) and for t    {I, 2, . . . ,} let x_(t)    Q⁺ and k(t)    {I, . . . , n}. Suppose for every t that k=k(t) satisfies (59) and    x _(t+1) ^(i) =x _(t) ^(i)(1+α(v _(t) ^(i) −x _(t) ^(k(t)))), i=1 .    . . , n.    where v_(t) ^(i)=1 if i=k(t) and v_(t) ^(i)=0 if i≠k(t). For all ε>0    let t_(ε) be the first t such that

$\begin{matrix}{{\frac{1}{1 + \varepsilon} \leq \frac{M_{t - 1}}{M_{t - 1} + 1} \leq \frac{M_{t - 1} + 1}{M_{t - 1}} \leq {1 + \varepsilon}}\mspace{14mu}{and}\mspace{14mu}{\frac{N_{t - 1}^{i} + 1}{N_{t}^{i} + 1} \geq {1 + {\varepsilon\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{i.}}}}} & (60)\end{matrix}$

-   (a) N_(t) ^(i)→∞ for every i. Hence for every ε>0 there is some t    after which (60) always holds.-   (b) There is a time T_(α) such that

$\begin{matrix}{{{{{\phi(\alpha)}q^{i}} - \left( {1 - {\phi(\alpha)}} \right)} \leq x_{t}^{i} \leq {{q^{i}/{\phi(\alpha)}}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} i\mspace{14mu}{and}\mspace{14mu} t} \geq T_{\alpha}}{where}{{\phi(\alpha)}:={\left( \frac{1}{1 + \alpha} \right)^{3}{\frac{\ln\left( {1 + \alpha} \right)}{\ln\left( {1/\left( {1 - \alpha} \right)} \right)}.}}}} & (61)\end{matrix}$By L'Hopital's rule φ(α)↑1 as α↓0.

-   PROOF: We know by Lemma 2 that M_(t)→∞ (with probability one).    Therefore to complete the proof of (a) it only needs to be shown    that N_(t) ^(i)→∞ for every i. By (59)

$\begin{matrix}{N_{t}^{k} = {{N_{t - 1}^{k} + 1} \leq {\left( \frac{M_{t - 1} + x_{t}^{k}}{M_{t - 1} + x_{t}^{i}} \right)\left( \frac{q^{k}/x_{t}^{k}}{q^{i}/x_{t}^{i}} \right)\left( {N_{t - 1}^{i} + 1} \right)} \leq {\left( {N_{t - 1}^{i} + 1} \right)\left( \frac{M_{t - 1} + 1}{M_{t - 1}} \right)\left( \frac{q^{k}}{q^{i}} \right)\left( \frac{x_{t}^{i}}{x_{t}^{k}} \right)\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{i.}}}} & (62)\end{matrix}$For any s and any outcome j we can write

$x_{t}^{j} = {{x_{s}^{j}{\prod\limits_{u = s}^{t - 1}\;{\left( {1 + {\alpha\left( {v_{u}^{j} - x_{u}^{k{(u)}}} \right)}} \right)\mspace{14mu}{for}\mspace{14mu} t}}} > {s.}}$But suppose there is an outcome i for which N_(t) ^(i) stops growing atsome time t=s, so that N_(t) ^(i)≦N_(s) ^(i) for all t. This means thatoutcome i is not bought for any t≧s, so v_(t) ^(i)=0 for t≧s. Then forthat i the above formula gives

$x_{t}^{i} = {{x_{s}^{i}{\prod\limits_{u = s}^{t - 1}\;{\left( {1 - {\alpha\; x_{u}^{k{(u)}}}} \right)\mspace{14mu}{for}\mspace{14mu} t}}} > {s.}}$Hence, for every t such that t≧s and t≧t_(ε), inequality (62) can bewritten as

${N_{t}^{k{(t)}} \leq {\left( {N_{s}^{i} + 1} \right)\left( \frac{M_{t - 1} + 1}{M_{t - 1}} \right)\left( \frac{q^{k{(t)}}}{q^{i}} \right)\left( \frac{x_{s}^{i}}{x_{s}^{k{(t)}}} \right){\prod\limits_{u = s}^{t - 1}\;\frac{1 + {\alpha\; x_{u}^{k{(u)}}}}{1 + {\alpha\left( {v_{u}^{k{(t)}} - x_{u}^{k{(u)}}} \right)}}}} \leq {\left( {N_{s}^{i} + 1} \right)\left( {1 + \varepsilon} \right)\left( \frac{q^{\max}}{q^{\min}} \right)\left( \frac{x_{s}^{\max}}{x_{s}^{\min}} \right)}},$where q^(max)=max_(j)q^(j), q^(min)=min_(j)q^(j), etc., and where thefact was used that each factor in the product is positive and bounded byone. This means that N_(t) ^(k) is bounded in t for any k that is everbought after t_(ε), which implies that N_(t) ^(i) is bounded for everyi. Since Σ_(j)N_(t) ^(j)=t→∞, this is a contradiction. Therefore N_(t)^(i)→∞ for every i and part (a) is proved.

To prove part (b) define the time-varying probabilities q_(t) ^(i)>0 by

$\begin{matrix}{q_{t}^{i} = {\frac{q^{i}/\left( {N_{t - 1}^{i} + 1} \right)}{\sum\limits_{j}{q^{j}/\left( {N_{t - 1}^{j} + 1} \right)}}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{i.}}} & (63)\end{matrix}$Then since the denominator is the same for all i condition (59) isequivalent to

$\begin{matrix}{{\left( {M_{t - 1} + x_{t}^{k}} \right)\frac{q_{t}^{k}}{x_{t}^{k}}} \geq {\left( {M_{t - 1} + x_{t}^{i}} \right)\frac{q_{t}^{i}}{x_{t}^{i}}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu}{i.}}} & (64)\end{matrix}$Since N_(t) ^(j)≧N_(t−1) ^(j) for all j.

$\begin{matrix}{q_{t + 1}^{i} = {{\left( \frac{N_{t - 1}^{i} + 1}{N_{t}^{i} + 1} \right)\left( \frac{\sum\limits_{j}{q^{j}/\left( {N_{t - 1}^{j} + 1} \right)}}{\sum\limits_{j}{q^{j}/\left( {N_{t}^{j} + 1} \right)}} \right)q_{t}^{i}} \geq {\left( \frac{N_{t - 1}^{i} + 1}{N_{t}^{i} + 1} \right)q_{t}^{i}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} i\mspace{14mu}{and}\mspace{14mu}{t.}}}} & (65)\end{matrix}$For i≠k we have N_(t) ^(i)=N_(t−1) ^(i) so (65) says q_(t+1) ^(i)≧q_(t)^(i). Since x_(t+1) ¹≦x_(t) ^(i) for such an i then

$\begin{matrix}{\frac{q_{t + 1}^{i}}{x_{t + 1}^{i}} \geq {\frac{q_{t}^{i}}{x_{t}^{i}}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} i} \neq {k\mspace{14mu}{and}\mspace{14mu}{all}\mspace{14mu}{t.}}} & (66)\end{matrix}$For i=k what we obtain from (65) is

$\begin{matrix}{\frac{q_{t + 1}^{k}}{x_{t + 1}^{k}} \geq {\left( \frac{N_{t - 1}^{k} + 1}{N_{t - 1}^{k} + 2} \right)\left( \frac{1}{1 + {\alpha\left( {1 - x_{t}^{k}} \right)}} \right){\frac{q_{t}^{k}}{x_{t}^{k}}.}}} & (67)\end{matrix}$As in (40), we have q_(t) ^(i)/x_(t) ^(i)≧1 for some i. Hence (64)implies that

$\frac{q_{t}^{k}}{x_{t}^{k}} \geq {\left( \frac{M_{t - 1} + x_{t}^{i}}{M_{t - 1} + x_{t}^{k}} \right)\frac{q_{t}^{i}}{x_{t}^{i}}} \geq \frac{M_{t - 1} + x_{t}^{i}}{M_{t - 1} + x_{t}^{k}}$for such an i and therefore by (60)

$\begin{matrix}{\frac{q_{t}^{k}}{x_{t}^{k}} \geq {\frac{1}{1 + \alpha}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} t} \geq {t_{\alpha}.}} & (68)\end{matrix}$By (60) and (68) and the fact that α(1−x_(t) ^(k))≦α when t≧t_(ε), weget from (67) that

$\begin{matrix}{\frac{q_{t + 1}^{k}}{x_{t + 1}^{k}} \geq {\left( \frac{1}{1 + \alpha} \right)^{3}\mspace{14mu}{for}\mspace{14mu} t} \geq {t_{\alpha}.}} & (69)\end{matrix}$Together, (66) and (69) imply that if for some t≧t_(α) we have

$\begin{matrix}{{\frac{q_{t}^{i}}{x_{t}^{i}} \geq {\left( \frac{1}{1 + \alpha} \right)^{3}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} i}},} & (70)\end{matrix}$then this will continue to be true for all subsequent t. Therefore for(70) to be false for all t≧t_(α) there would have to be some i for whichq_(t) ^(i)/x_(t) ^(i)<1/(1+α)³ for all t≧t_(α) but by (68) such an iwould never be bought after t_(α) and that would contradict N_(t)^(i)→∞. We conclude that (70) must be valid for some t≧t_(α) and hencefor all t≧t_(α).

So far it has been shown that every q_(t) ^(i)/x_(t) ^(i) converges to asmall neighborhood of 1. To complete the proof we will show that q_(t)^(i) converges to a small neighborhood of q^(i). By (63)

$\begin{matrix}{q^{i} = {q_{t}^{i}{\sum\limits_{i}{{q^{j}\left( {N_{t - 1}^{i} + 1} \right)}/{\left( {N_{t - 1}^{j} + 1} \right).}}}}} & (71)\end{matrix}$We will use Lemma 3 to show that

$\begin{matrix}{{{\underset{t\rightarrow\infty}{\lim\mspace{14mu}\inf}\frac{N_{t - 1}^{i} + 1}{N_{t - 1}^{j} + 1}} \geq {\frac{\ln\left( {1 + \alpha} \right)}{\ln\left( {1/\left( {1 - \alpha} \right)} \right)}\mspace{14mu}{for}\mspace{14mu}{any}\mspace{14mu} i}},{j.}} & (72)\end{matrix}$Putting (70), (71) and (72) together gives

${{\underset{t\rightarrow\infty}{\lim\mspace{14mu}\inf}\frac{q^{i}}{x_{t}^{i}}} \geq {\left( \frac{1}{1 + \alpha} \right)^{3}\frac{\ln\left( {1 + \alpha} \right)}{\ln\left( {1/\left( {1 - \alpha} \right)} \right)}}} = {{\phi(\alpha)}.}$This implies part (b) of the theorem: the right inequality in (61) isimmediate and the left inequality comes from

$x_{t}^{i} = {{{1 - {\sum\limits_{j \neq i}x_{t}^{i}}} \geq {1 - {\sum\limits_{j \neq i}{q^{j}{\phi(\alpha)}}}}} = {{1 - {{\phi(\alpha)}\left( {1 - q^{j}} \right)}} \geq {{{\phi(\alpha)}q^{j}} - {\left( {1 - {\phi(\alpha)}} \right).}}}}$To prove (72) define

$a_{t}^{i} = {{\left( \frac{M_{t - 1} + x_{t}^{i}}{M_{t}} \right){\left( \frac{q_{t}^{i}}{x_{t}^{i}} \right).b_{t}^{i}}} = {\left( \frac{M_{t} + x_{t + 1}^{i}}{M_{t - 1} + x_{t}^{i}} \right)\left( \frac{N_{t - 1}^{i} + 1}{N_{t - 1}^{i} + 2} \right)\left( \frac{\sum\limits_{j}{q^{j}/\left( {N_{t - 1}^{j} + 1} \right)}}{\sum\limits_{j}{q^{j}/\left( {N_{t}^{j} + 1} \right)}} \right)}}$Then k(t) is an i that maximizes a_(t) ^(i) and we can write theupdating of a_(t) ^(i) as

${a_{t + 1}^{i} = \frac{a_{t}^{i}b_{t}^{i}}{1 + {\alpha\left( {v_{t}^{i} - x_{t}^{k{(t)}}} \right)}}},$as one can confirm by inspecting the expanded version

${\left( \frac{M_{t} + x_{t + 1}^{i}}{M_{t}} \right)\left( \frac{q_{t + 1}^{i}}{x_{t + 1}^{i}} \right)} = {\left\lbrack {\left( \frac{M_{t - 1} + x_{t}^{i}}{M_{t}} \right)\left( \frac{q_{t}^{i}}{x_{t}^{i}} \right)} \right\rbrack\left( \frac{M_{t} + x_{t + 1}^{i}}{M_{t - 1} + x_{t}^{i}} \right){\left( \frac{N_{t - 1}^{k} + 1}{N_{t - 1}^{i} + 2} \right) \cdot \left( \frac{\sum\limits_{j}{q^{j}/\left( {N_{t - 1}^{j} + 1} \right)}}{\sum\limits_{j}{q^{j}/\left( {N_{t}^{j} + 1} \right)}} \right)}{\left( \frac{1}{1 + {\alpha\left( {v_{t}^{i} - x_{t}^{k{(t)}}} \right)}} \right).}}$It is straight-forward to verify that b_(t) ^(i)→1 with probability one.Hence, applying Lemma 3 from the appendix with the a_(t) ^(i) and b_(t)^(i) there representing the reciprocals of the above a_(t) ^(i) andb_(t) ^(i), one obtains (72) and that completes the proof.17.7 References used in this section include Karatzas, I. and S. E.Shreve, (1991) Brownian Motion and Stochastic Calculus, 2^(nd) Edition,New York: Springer-Verlag. and Kushner, Harold J. and George, Yin G.(2003) Stochastic Approximations and Recursive Algorithms andApplications. New York: Springer-Verlag.17.8 Appendix: LemmasAll the lemmas used in the section are in this appendix. Lemma 2, below,is by itself an interesting property of the algorithm that isindependent of how investors behave.

-   LEMMA 1: Let u_(t) and α_(t), t≧1, be sequences of numbers such that    u_(t)>0, α_(t)≧ and u_(t+1)≧u_(t)(1−αt) for all t. If Σ_(t)≧1    α_(t)<∞ then there is some b>0 and some T≧1 such that u_(t)≧b for    every t≧T.-   PROOF: Without loss of generality, we can assume that    Σ_(t≧1)α_(t)<1, for otherwise we can consider the sequences Ut and    at for large enough t only. Just like we have    (1−a _(t))(1−a _(t+1))=(1−(a _(t) +a _(t+1))+a _(t) a _(i+1))≧1−(a    _(t) +a _(t+1)),    one can show by induction that

${\prod\limits_{s = 1}^{t}\;\left( {1 - a_{s}} \right)} \geq {1 - {\sum\limits_{s = 1}^{t}{a_{s}.}}}$Therefore, writing b=u₁(1−Σ_(s≧1)α_(s))>0, we have for t≧2

$u_{t} = {{u_{1}{\prod\limits_{s = 1}^{t - 1}\;\frac{u_{s} + 1}{u_{s}}}} = {{{u_{1}{\prod\limits_{s = 1}^{t - 1}\;\left( {1 - a_{s}} \right)}} \geq {u_{1}\left( {1 - {\sum\limits_{s = 1}^{t - 1}a_{s}}} \right)} \geq {u_{1}\left( {1 - {\sum\limits_{s = 1}^{\infty}a_{s}}} \right)}} = {b.}}}$

-   LEMMA 2: For each tε{1, 2 . . . . } and iε{1, . . . , n} let x_(t)    ^(i)ε(0,1), α_(t)ε(0,1) and k(t)ε{1, . . . , n}. Assume that    x _(t+1) ^(i) =x _(t) ^(i)(1+α_(t)(v _(t) ^(i) −x _(t) ^(k(t)))) for    i=1, . . . , n,  (73)    where v_(t) ^(i)=1 if i=k(t) and v_(t) ^(i)=0 if i≠k(t). Then    Σ_(t≧1)x_(t) ^(k(t))=∞. If also Σ_(t≧1)α_(t)=∞, then    Σ_(t≧1)α_(t)x_(t) ^(k(t))=∞ (Since (37) can also be written as    x_(t+1) ^(i)=(1−β_(t)) x_(t) ^(i)+β_(t)v_(t) ^(i), where    β_(t)=α_(t)x_(t) ^(k(t)), it is clear that x_(t) ^(i)ε(0, 1) for all    t if x₁ ^(i)ε(0, 1).-   PROOF: Define x _(t)=min_(j)x_(t) ^(j). Then for all i we have    x_(t+1) ^(i)≧x_(t) ^(i)(1−α_(t)x_(t) ^(k(t)))≧ x _(t)(1−α_(t)x_(t)    ^(k(t))), so    x _(t+1) ≧ x _(t)(1−α_(t) x _(t) ^(k(t))).  (74)    If Σ_(t≧1)x_(t) ^(k(t))<∞ then Σ_(t≧1)α_(t)x_(t) ^(k(t))<∞, so    by (74) and Lemma 1, x_(t) ^(k(t))≧ x _(t)≧b>0 for t≧T, for some b>0    and some T≧1. The conclusions of the Lemma follow from this.-   LEMMA 3: For tε{1, 2, . . . } and iε{1, . . . , n}, let x_(t) ^(i),    a_(t) ^(i), b_(t) ^(i)>0 and k(t)ε{1, . . . , n}. Let αε(0,1).    Assume that lim_(t→∞)b_(t) ^(i)=1, k(t)εarg min_(j)a_(t) ^(j) and

${a_{t + 1}^{i} = {a_{t}^{i}{b_{t}^{i}\left( {1 + {\alpha\left( {v_{t}^{i} - x_{t}^{k{(t)}}} \right)}} \right)}}},{{{where}\mspace{14mu} v_{t}^{i}} = \left\{ \begin{matrix}{1,} & {{i = {k(t)}},} \\{0,} & {i \neq {{k(t)}.}}\end{matrix} \right.}$Let N_(t) ^(i) be the number of sε{1 . . . , t} such that k(s)=i. Then

$\begin{matrix}{{\frac{\ln\left( {1 + \alpha} \right)}{\ln\left( {1/\left( {1 - \alpha} \right)} \right)} \leq {\underset{t\rightarrow\infty}{\lim\mspace{14mu}\inf}\frac{N_{t}^{i}}{N_{t}^{j}}} \leq {\underset{t\rightarrow\infty}{\lim\mspace{14mu}\sup}\frac{N_{t}^{i}}{N_{t}^{j}}} \leq {\frac{\ln\left( {1/\left( {1 - \alpha} \right)} \right)}{\ln\left( {1 + \alpha} \right)}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} i}},{j.}} & (75)\end{matrix}$The ratio on the far left increases to one and the ratio on the farright decreases to one as a↓0.

-   PROOF: Define c_(t) ^(i) by

${\exp\left( c_{t}^{i} \right)} = {a_{t}^{i}{\prod\limits_{u = 1}^{t - 1}\;{\frac{1}{1 - {\alpha\; x_{u}^{k{(u)}}}}.}}}$Then since the product does not depend on i, for every t these numbersare proportional to the a_(t) ^(i) and hencearg min_(j) a _(t) ^(j)=arg min_(j)exp(c _(t) ^(j))=arg min_(j) c _(t)^(j),so k(t) can also be chosen by minimizing over the c_(t) ^(i). We willwork with the c_(t) ^(i) because the updating rule for them is simpler:the smallest (or rather a smallest) c_(t) ^(i), say c_(t) ^(k), isincreased to

$\begin{matrix}{c_{t + 1}^{k} = {c_{t}^{k} + {\ln\; b_{t}^{k}} + {{\ln\left( \frac{1 + {\alpha\left( {1 - x_{t}^{k}} \right)}}{1 - {\alpha\; x_{t}^{k}}} \right)}.}}} & (76)\end{matrix}$and the other c_(t) ^(i) are left as they were, c_(t+1) ^(i)=c_(t) ^(i).

Assume for now that b_(t) ^(i)=1 for every i and every t. Then theincrement to c_(t) ^(k(t)) lies in the interval

$\begin{matrix}{\left( {{\ln\left( {1 + \alpha} \right)},{\ln\left( \frac{1}{1 - \alpha} \right)}} \right).} & (77)\end{matrix}$Note from the lower bound that c_(t) ^(i)→∞. We will prove the rightmostinequality in (75), which by itself implies the leftmost inequality.

Since at every t a smallest c_(t) ^(i) is raised by a number that isbounded above by r:=ln(1/(1−a)), one can show that there is some time ssuch thatc _(t) ^(max) −c _(t) ^(min) ≦r for all t≧s.  (78)where c_(t) ^(max)=max_(j)c_(t) ^(j) and c_(t) ^(min)=min_(j)c_(t) ^(j).For example

${s = {\min\left\{ {{{t \geq 1}:{k(t)}} = {{i\mspace{14mu}{for}\mspace{14mu}{some}\mspace{14mu} i} \in {\arg{\mspace{11mu}\;}{\min\limits_{j}c_{1}^{j}}}}} \right\}}},$since if c₁ ^(i) is maximal then c_(t) ^(i) will sit at c₁ ^(i) untilevery other c_(t) ^(j) catches up with it or passes it, and when they dothat they land within r of c₁ ^(i). Similarly c_(t) ^(j)≦c_(t) ^(k(t))+rfor all j for all t after that.

Define

$\begin{matrix}{v_{s,t}^{+} = {{\frac{c_{t}^{\max} - c_{s}^{\min}}{\ln\left( {1 + \alpha} \right)}\mspace{14mu}{and}\mspace{14mu} v_{s,t}^{-}} = {\frac{c_{t}^{\min} - c_{s}^{\max}}{\ln\left( {1/\left( {1 - \alpha} \right)} \right)}.}}} & (79)\end{matrix}$To go from c_(s) ^(i) to c_(t) ^(i) it must be the case that k(u)=i forsome uε{s, s+1, . . . , t−1} somewhere between v_(s,t) ⁻ and v_(s,t) ⁺times. Hence one can say thatv _(s,t) ⁻ ≦N _(t) ^(i) −N _(s) ^(i) ≦v _(s,t) ⁺, i=1, . . . , n, t≧s.Then, for any i and j, since N_(t) ^(i), N_(t) ^(j)→∞,

$\begin{matrix}{{\underset{t\rightarrow\infty}{\lim\mspace{14mu}\sup}\frac{N_{t}^{i}}{N_{t}^{j}}} = {{\underset{t\rightarrow\infty}{\lim\mspace{14mu}\sup}\frac{N_{t}^{i} - N_{s}^{i}}{N_{t}^{j} - N_{s}^{j}}} \leq {\underset{t\rightarrow\infty}{\lim\mspace{14mu}\sup}{\frac{v_{s,t}^{+}}{v_{s,t}^{-}}.}}}} & (80)\end{matrix}$Note from (79) that v_(s,t) ⁻ can be negative if t is too close to s,but it becomes positive once t increases away from s, so the last limitin (80) is well-defined.

We will now develop bounds for v_(s,t) ^(±). If v_(s,t) ⁻>0,

$\begin{matrix}\left. {1 \leq \frac{v_{s,t}^{+}}{v_{s,t}^{-}} \leq {\left( \frac{c_{t}^{\max} - c_{s}^{\min}}{c_{t}^{\min} - c_{s}^{\max}} \right)\frac{\ln\left( {1/\left( {1 - \varepsilon} \right)} \right)}{\ln\left( {1 + \varepsilon} \right)}}}\rightarrow\left. {\frac{\ln\left( {1/\left( {1 - \alpha} \right)} \right)}{\ln\left( {1 + \alpha} \right)}\mspace{14mu}{as}\mspace{14mu} t}\rightarrow{\infty.} \right. \right. & (81)\end{matrix}$since by (78)

$\begin{matrix}{\frac{c_{t}^{\max} - c_{s}^{\min}}{c_{t}^{\min} - c_{s}^{\max}} = \left. {{1 + \frac{c_{t}^{\max} - c_{t}^{\min} + c_{s}^{\max} - c_{s}^{\min}}{c_{t}^{\min} - c_{s}^{\max}}} \leq {1 + \frac{2\; r}{c_{t}^{\min} - c_{s}^{\max}}}}\rightarrow 1. \right.} & (82)\end{matrix}$By (80) this shows that (75) is true for the case where b_(t) ^(i)=1.

When instead of b_(t) ^(i)=1 we have b_(t) ^(i)→1, the same proof workswith the following changes. In the increment to c_(t) ^(k(t)) shown in(76) the term b_(t) ^(k) now may be nonzero but tends to zero. Thebounds in (77) are therefore enlarged on each side by an arbitrarilysmall δ>0 and s is increased to a point such that for all t≧s theenlarged interval contains the increment. Then the denominators in (79)are incremented by δ, so in the end instead of (the last inequality in)(75) we obtain

${{\underset{t\rightarrow\infty}{\lim\mspace{14mu}\sup}\frac{N_{t}^{i}}{N_{t}^{j}}} \leq {\frac{{\ln\left( {1/\left( {1 - \alpha} \right)} \right)} + \delta}{{\ln\left( {1 + \alpha} \right)} + \delta}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} i}},{j.}$Thus this holds for all small δ>0 and therefore it holds also for δ=0,which proves the third inequality in (75). The lemma is now proved.

Referring now to FIG. 49, a schematic illustration of a flow chart forcalculating pricing probabilities and hence market pricing, is shown.According to the method described herein and as outlined in the flowdiagram of FIG. 49, pricing probabilities π are computed for each of thegeographical units of interest with each purchase requested. Therequests to purchase may be expressed in terms of purchasing shares,commodity options or contracts or other financial investment units. Aswill be seen herein, the flow chart of FIG. 49 is restricted to requestsfor a particular geographical unit, such as a county, designated by thereference character k in this section and elsewhere. The requests topurchase may however be made for geographical units of interest. Therequest currently under consideration is acquired in block 780,preferably using any of the network arrangements described herein,populated by digital programmable computers or other communicationdevices as considered herein. As indicated in the upper right corner ofFIG. 49, the current Request is taken from a series of requests R₁, R₂,R₃ and so forth. Accordingly, for example, if a particular individualwished to present a menu of purchase requests for the same or differentgeographical units, requests R₁ through R₃, for example, may all be madeby the same individual, with each request being considered sequentially.Further, as will be seen herein, multiple options for a particulargeographical unit are also processed sequentially with a calculation ofpricing probability and hence market price being calculated with aniteration for each option requested to be purchased.

In the flow diagram of FIG. 49 and in the formulas contained herein, thenumber of options for a particular geographical unit requested, isidentified by the reference character n. Block 782 indicates that thefinancial activity has received a request by the user to purchase noptions in an outcome k, herein a geographical unit offered toparticipants in the financial activity. In the preferred First LandfallMarket embodiment, the graphical user interfaces described herein areemployed to present a map of geographical units, k, such as counties, tothe participants. Preferably, the pricing probabilities and/or themarket prices for the geographical units are also displayed on the samemap for ready reference by a participant.

In block 784 the reference character i indicates an internal counterwhich runs from I to the total number n, of options requested. In block784, initially, the first option is considered with i=1. The price forthe first option is recorded according to the formula expressed in block784. Next, the pricing probability for the geographical unit of interestis increased according to equation 85.

Control is then transferred to block 796 where pricing probabilities forall other geographical units (i.e., all geographical units other thangeographical unit k) are updated, preferably according to equation 86cited in this section. Generally speaking, pricing probability for thegeographical unit in which the current purchase has been confirmed isincremented or increased according to equation 85, while the pricingprobabilities for all other geographical units are lowered according toequation 86 in this section. For each pricing probability, acorresponding market price can be computed using equation 83.

In block 788, control is transferred back to block 784 and the counter iis incremented to record the price for the second option requested.Control is continuously transferred by block 788 until all of thepricing probabilities have been calculated. In block 786 the term“provisional” indicates that the current pricing probability iscalculated but not yet entered into the system, pending acceptance bythe offerer and participant of the financial activity.

When all of the provisional pricing probabilities have been calculated,the total price for all of the requested options is computed in block790. In block 792 all necessary confirmations of the investment areobtained. This may include, for example, a confirmation by theparticipant that the total price is agreeable, and confirmation may alsohave been made by the provider of the financial activity that theparticipant's credit is satisfactory. When all confirmations have beenobtained, control is transferred to block 794 which indicates that theprovisional pricing probabilities have been accepted and are accountedfor in the system as a basis for future activity. In block 794 the primesuperscript is removed from the pricing probability to indicate that thepricing probability is no longer provisional, but has been accepted inthe system.

Control is then transferred to an optional block 804 that reports thepricing information to the graphical user interface, to inform the nextparticipant of the updated prices. Preferably, the graphical userinterface of FIG. 24 and following is employed to display the pricinginformation in the form of a map of geographical units (outcomes), mostpreferably as a display of interactive elements by which theparticipants communicate with the financial activity administrator.

In the flow diagram of FIG. 49, control is transferred from block 804 toblock 798 where the current price for the current option underconsideration is obtained and presented to the requester. If desired,control can be passed to block 798 from block 786 since the methodoutlined in the flow diagram of FIG. 49 assumes a constant selection ofgeographic location. In any event, for a new participant, the requestmay be for a different outcome, and accordingly, updating of theremaining outcomes should be completed in block 796 before supplyingcurrent price to the new participant.

If desired, when a request for a block of purchases for the sameoutcome, control can be passed from block 790 to block 798 when thetotal price for the n options has been determined and the intermediatecalculated prices calculated as counter i progresses from 1 to n, may beomitted if it is assumed that the participant's interest is in the totalprice for the request currently under consideration.

Referring to the upper right hand corner of FIG. 49, at the initialoperation of the financial activity a current price based on marketactivity will not be available for presentation to the requester, sincemarket activity has not yet begun. For this initial transaction, controlis transferred to block 800 which calculates an initial pricingprobability based, for example, on historical values. Block 800 reflectsthe geographical units considered in the graphical user interfaceexamples herein, wherein 78 geographic units are presented to theparticipants, along with a “null” event. When the transaction iscompleted, control is passed to block 802 which in turn transferscontrol to block 782 along with incrementing the request in block 780from the next requestor in the queue. As mentioned, the next request maybe made by the same individual, but for a different geographical unit k,or may be made by a different individual with another investment choicein mind.

As indicated above, it is generally preferred that a graphical userinterface, such as those presented herein with reference to FIG. 24 andfollowing, are employed to conduct the financial activity. The mapsemployed, in addition to identifying the geographical boundaries of eachlocation available for investment, indicate a current market price (or acorresponding price probability) for each geographical location andoptionally a running counter for each of the geographical units madeavailable purchaser, which reflects the number of previous purchases forthat geographic location. As mentioned herein, it is generally preferredthat the payout be made on a pari-mutuel basis and accordingly it isadvantageous to provide an indication of the current total of moniesavailable for payout. It is generally preferred that each of theseparameters be stored in one or more data tables for ready reference by acomputer program that implements the graphical user interface andconducts the financial transaction associated therewith. In addition tothe description of each geographical unit available in the financialactivity, a data table may be employed to store an identification of thegeographical unit by either name or number, made available for readyreference to expedite the participant's requests. Data tables may alsobe employed to store any of the current relative probability of beingthe next successful outcome/pricing probability/market price for eachgeographic unit in play as well as the number of options purchased foreach geographical unit. As mentioned, it is generally preferred that thepricing probability/market price be proportional, or equal to, thecurrent relative probability of a particular outcome (geographical unit)being the next successful outcome.

A data table may also be maintained to indicate the amount of moniescurrently available for payout. Each of these databases are preferablymaintained for ready access by a processor, programmable or dedicatedcomputer or other device as described herein, for the automated conductof financial activities according to principals of the presentinvention. The pricing probabilities (and/or market prices) are updatedin a manner described herein with reference to the flow chart of FIG.49, for example. Market prices in the data tables may be calculated fromthe pricing probabilities using equation 83, for example.

XVIII. CONSIDERATIONS REGARDING PAYOUT

In general, the basis or methodology of the payout can be eitherpari-mutuel or non pari-mutuel (e.g. a fixed or a varying payout price,such as one based on a scale or index). As a further alternative, payoutcan be structured on a modified pari-mutuel basis. For example, a poolof money can be split (after recovering overhead costs) among thequalifying participants in an unequal, i.e. modified manner, depending,in one instance, on an algorithm, or on a scale or index or in anotherinstance, on a requirement to fund an expected minimum payout, or“floor,” as will be discussed herein.

If desired, financial activities relating to hurricane natural perilevents can be structured around hurricane landfall or hurricane landtracks, either individually, one exclusive of the other, or incombination. A question arises, for example when a hurricane landfall ismade on or close to a border between two unit areas (i.e. geographicalareas, such as counties used to define purchase units). An arbitrarywidth can be assigned to the point of landfall if desired, by theprovider of the financial activity.

The passage of a hurricane over inland geographical areas can raise anumber of different possibilities made available to the provider of afinancial activity. For example, if the National Hurricane Center isdesignated as the external objective independent information source, onereport currently available to providers of financial activities is theso-called “best track” report which issues after a hurricane event isconcluded. The “best track” report defines the inland path of ahurricane according to a table of discrete position values. Thus, it isleft to the provider of the financial activity to determine the best wayto define the hurricane path between published points on the “besttrack” table. If desired, the points of the “best track” table can beconnected by straight lines or by curved lines according to a predefinedcurve-fitting method, for example. As a further possibility, anarbitrary width can be assigned to the hurricane path.

If desired, other sources of information can be employed since the “besttrack” report is not the only possible source of scoring information,and may not be desirable in certain instances because of the time delayassociated with issuance of the report after conclusion of the naturalactivity. For example, the same kind of information (lat/lon and maximumsustained winds) are available in near-real time in what are called“advisories” issued by the National Hurricane Center, which do notsuffer from prolonged time delays. Preferably, the most rapidsatisfactory resolution of the outcome of the natural peril event ispreferred, so that distributions can be made promptly to individuals whosuffer from the natural peril event.

For other types of natural activity, such as tornadoes, the path and/orintensity can be reported by an official source, and for earthquakes,the path of damage and intensity can be specified according to anindependent third party.

Different possibilities are presented when considering a particulargeographical land unit. For example, payout for a geographical land unitcan be based upon one or more external factors, such as a simplehit/no-hit treatment for the land unit of interest. In another example,payout for a land unit of interest can be based upon the publishedstrength of the hurricane according to the “best track” or other table.As a further possibility, it is recognized that the strength of ahurricane can vary in intensity or strength when passing over a givengeographical land unit of interest. The possibility is thus presentedfor a mathematical treatment taking into account the difference ofstrengths at entry and exit points of the hurricane with respect to thegeographical unit of interest. As a further possibility, thegeographical unit of interest can lie between points published on the“best track” table, and some manner of interpolation of values can bemade with respect to the geographical unit of interest. If desired,payouts can be calculated based upon the strength of the hurricane forthe qualifying geographical unit. For example, one payout possibility isto award greater payout for geographical units suffering greaterstrengths of hurricane activity, under the premise that more help willbe provided for those participants that suffer greater damage, asmeasured by hurricane strength.

Variability factors other than those presented above can also beconsidered when calculating payouts to qualifying participants. Forexample, in addition to timing factors and probability factors discussedabove (which are preferably employed for price variability as well aspayout variability) an account can be made of the residence time of ahurricane in a given geographical unit of interest, under the premiseagain that more help will be provided for those participants that suffergreater damage as measured by the time that a hurricane is present in agiven geographical unit of interest.

It is sometimes preferred to provide an expectation of a minimum payoutor “floor” for participants that suffer damage from a hurricane or othernatural peril event. The “floor” is a minimum payout, conditional on thechosen geographical location (in which a market participant holds aninvestment unit) being “hit” (i.e. suffers a predefined amount or typeof contact with a storm).

In one example, the floor is the same fixed dollar value for allfinancial investment units in a given county (or other geographicalregion), and for all counties (all such regions). In one example, it iscomputed in the following way: One parameter in the pricing algorithm isprovided as the “par” value. Prices are computed as the product of theprobability (as assessed at any given time for the county or region inquestion) multiplied by the par value (and also multiplied by atime-value-of-money escalator). The par value is called “par” in thisexample, because, if the market is working smoothly, the payout a marketparticipant might expect is at or near the par level.

In instances where there is very strong buying activity, the actualpayout may fall below the par level. If it falls far enough below, thefloor is triggered. The floor, in one example, is a fixed percentage(specific value to be determined as a percentage of the par. For thefollowing example: it is assumed that a probability of the selectedcounty (or region) being hit, as assessed at the time of purchase, is5%, and that par value has been set at $1000. Neglecting thetime-value-of-money escalator, $50 is paid for one financial investmentunit. If the market works smoothly, the eventual payout, should theselected county be first hit, will be in the neighborhood of $1000. Butif market circumstances (e.g., very strong buying interest just beforelandfall) pushes the actual payout low enough, an expectation is, in theexample, given that a payout of least $800, will be received if thefloor has been set at 80%.

A mechanism is needed to honor the floor. One possibility is that, if anew purchase would drive the current indicated payout (total in anassumed pari-mutuel, or mutual risk pool divided by number of financialinvestment units for this county or region) below the floor level, theprice for that new purchase suddenly spikes to the floor level. In mostcases this will dissuade the would-be purchaser from buying in a primarymarket, and motivate that person to seek a better price in a secondarymarket, if one is made available.

Consideration is also given to the possibility of a payout that addressthe timing of the payout, or conditions precedent to a settlement of thefinancial activity.

If desired, the payout can employ a sliding scale, based on any numberof factors that may be in play for a given financial activity.

XIX. MODULARITY

In one instance, one or more systems, one or more methods, and/or one ormore devices for carrying out the financial activity are provided. Anumber of important issues are addressed by the databases and/orprogram. In one instance, it is a generally preferred that these issuesbe addressed as much as possible, on a modular basis. In this manner, asystem administrator is able to quickly and easily tailor the financialactivity to meet a number of particular needs, and can modify thefinancial activity on an ongoing basis, if necessary. A briefdescription of some of the “modular” issues will now be given.

19.1 Definition of an external objective independent agency whichmonitors a natural peril event, measures, observes, or otherwise obtainsand records data concerning a natural peril event, as well as drawingconclusions and making analytical determinations concerning a naturalperil event. In one instance, it is generally preferred that theexternal objective independent agency be independent of theparticipant's financial activity, and in another instance be readilyobservable by the public, or at least the participants. For example, theexternal objective independent agency can comprise a unit of the UnitedStates government which routinely makes public announcements and whichis subject to Freedom of Information inquiries from members of thepublic.

19.2 Definition of an event eligible for payout. For example, relatingto hurricane natural peril events, it is generally preferred in oneinstance that the event be “officially” declared a “hurricane” asdefined by the National Hurricane Center. However, in other instances,other recognized pre-hurricane stages can be treated by the financialactivity, with or without weighting the points upon which payout isbased. In one instance, payout for the financial activity can be basedupon an occurrence of a natural peril event or a termination of anatural peril event.

19.3 Definition of participant eligibility needed to be permitted toengage in the financial activity. Included, for example, is the level ofskill of the participant (needed, to qualify to participate in afinancial activity structured as a game of skill), or the propertyrights of participant (needed, for example, to qualify to participate ina financial activity structured as a vehicle for recouping losses toproperty rights caused by a natural peril event). It may be desirable,in certain instances to have an outside party handle these types ofactivities. For example there are a number of known enterprises thatassess the financial responsibility of individuals and businesses. Thisactivity may or may not be combined with an outside party that handlescash transfers and related matters.

19.4 Definition of a “season” for the financial activity. The financialactivity season can, for example, coincide with a particular timeinterval such as a “hurricane season” as defined by the NationalHurricane Center. In another instance, the financial activity season canbe chosen to lie outside of a recognized or customary time period suchas the National Hurricane Center “hurricane season” and that this ispreferred for hurricane natural peril events.

19.5 Number and length of financial activity seasons in a given year. Inone instance, there can be but one financial activity season. In oneinstance, the financial activity season can begin at the beginning of acalendar year. In another instance, the financial activity season canbegin at any time during a calendar year. In one instance, the length ofa financial activity season can be a predefined number of natural perilevents. In another instance, a financial activity season can be definedto comprise a predetermined number of natural peril events, which iseither concluded or is followed by a subsequent financial activityseason upon the occurrence of those predetermined number of naturalperil events.

19.6 Defining the types of natural peril events and activities uponwhich payouts are based. For example, for hurricane events, recognizedactivities can include coastal strikes, inland strikes and near-shorehurricanes which do not make landfall (such as hurricanes which comewithin one quarter mile of eligible coastal shore). Other definitions of“sub-characteristics can be made for other types of natural perilevents, other than hurricanes.

19.7 Defining areas or regions eligible for inclusion in the financialactivity. In one instance, only terrestrial areas or regions may bedeclared eligible for inclusion in the financial activity. In anotherinstance, the terrestrial areas or regions eligible for inclusion in thefinancial activity are geographically defined according to convenientdelineations, such as established political boundaries. In a furtherinstance, portions of geographic regions can be declared ineligible forinclusion (for example, some of the many island areas of the EasternSeaboard of the United States can be declared ineligible for inclusion,because of small size, few or no inhabitants, or for other reasons whichare or are not stated).

19.8 Defining the areas stricken by a natural peril event. In oneinstance, the stricken areas can be defined according to externalobjective independent agencies such as the National Hurricane Center. Inone example, stricken areas eligible for the financial activity includethose areas as defined by the National Hurricane Center “best-track” orother, interim reports which are typically published either during orshortly after the conclusion of a hurricane event. In one instance, thesize or width of the National Hurricane Center “best track” (preferably,of the center of the eye of the hurricane) can be infinitely thin, or itcan be of a predetermined width. In one instance, stricken areaseligible for the financial activity can be calculated by connectingpoints given in the National Hurricane Center table data of ahurricane's “best-track” or other table with either a straight line, ora curved line preferably defined by a predetermined curve-fittingmethod.

19.9 Defining the nature of the natural peril event to be eligible forthe financial activity. For example, events officially determined to bewell-defined “hurricanes” by the National Hurricane Center can bedeclared by the rules of operation as the only eligible natural perilevent recognized by the financial activity. In another instance, thefinancial activity can treat “hurricanes” defined by the NationalHurricane Center according to their storm intensity as defined by theNational Hurricane Center. For example, only a hurricane defined asreaching category three severity by the National Hurricane Center can bedeclared eligible for the financial activity. In one instance, payoutscan be based upon hurricane strikes, weighted according to their stormintensity as defined by the National Hurricane Center. For example,distributions based upon a hurricane's “best track” can pay outdifferent amounts for different qualifying participants, depending uponthe severity of the hurricane at the time and/or point of contact withthe hurricane, or other primary, secondary, tertiary or other criteria.

In one instance, observed information from an independent externalsource regarding the land track of a tropical weather event may not becontinuously reported. For example, the use of a “best track” or othertable inherently assumes discrete points of data spread out over a timeinterval. Questions can arise when the reported data does not correspondto boundaries of geographical areas defined by the financial activity.Various treatments can be given. For example, an average value can beestablished between two adjacent data points (e.g. two adjacent pointsof a “best track” or other table) and this average value can be used todetermine the value of the natural peril event as it passed through agiven geographical area. In another treatment, if a data point (e.g. apoint on the “best track” or other table) occurs within a geographicalarea of financial activity, the value attributed to the data point canbe used for all investments made within the geographical area predictedby a participant. Other treatments are also possible.

19.10 Determining the amount of payout for those participants eligibleto receive payouts, as well as the conditions that must be present for apayout or other settlement to occur. Several instances of payouts variedaccording to a number of different factors and considerations are givenherein. These variations can be accounted for in a number of differentways including, for example, a simple linear weighting or a more complexalgorithm, formula, index or table, based upon historical events, orobserved natural peril events, for example. In another instance,variations in payout between different participants can be based uponone or more related or independent factors, as may be desired. In oneinstance, determining the amount of payout for qualifying participantscan be based upon primary, secondary and if desired, tertiary and othercriteria. For example, for tropical weather events, primary criteria canbe chosen to be the “locus” of landfall of a tropical weather event. Ifdesired, the land track of a tropical weather event can be treated asanother primary criterion (especially where an equal weighting amongprimary criteria is assumed) or can be treated as a secondary criterion(especially where unequal, preferably a lesser, weighting is assigned,relative to the primary criteria). In another instance, other secondaryor lower level criteria can be chosen, such as residence time in a givengeographical area, or wind speed or range of wind speed associated witha tropical weather event, preferably while the tropical weather event isresident in the geographical area predicted by the participant. As afurther possibility, multiple criteria can be established in tertiary orother additional levels (preferably assuming unequal weighting among thelevels of the criteria).

Other examples of variability factors are discussed herein.

XX. Derivative Trading Financial Activity Model

As mentioned above, financial activities may, in one instance, bemodeled to include or resemble financial trading of derivativesecurities interests (e.g. options), such as those monitored by theCommodity Futures Trading Commission (an independent agency of theUnited States government), the New York Stock Exchange, the ChicagoMercantile Exchange, the Iowa Electronic Market, and others. Thesefinancial activities may include, for example, futures contracts,options contracts, options on futures contracts and other forms ofderivative products. These are the preferred types of activity forpracticing the First Landfall Market, considered herein.

Generally speaking, compared to other types of financial activitiesconsidered herein, activities under the “derivative trading type”financial model may, in one instance, incorporate price controls drivenmore by market conditions and less by direct control via the rules,principles of operation and other structures of the financial activitybeing undertaken. If desired, the financial activity may be entirelymarket driven.

According to one example of this financial activity model, and assuminga hurricane or tropical weather type of natural peril event, initiallyall geographical areas (e.g. counties) in play are available forindividual investments at the beginning of the declared financialactivity season at some predefined minimum investment amount (i.e.purchase price). One example of a derivative trading financial activitycomprising a vehicle for the First Landfall Market, is referred toherein as the FIRST LANDFALL OPTIONS, or FLO. As mentioned herein, theprinciples set out in the following and other examples, may be readilyapplied to other types of natural peril events. For instance, thefollowing first landfall options could be readily recast as a firsttornado or earthquake strike options based on geographically definedareas of these types of natural peril events.

20.1 First Landfall Options, Or FLO

a. Background. On average for the last 130 years, two hurricanes makelandfall in the U.S. each year. The activity will assume there are 78individual counties, parishes or other land units on the Atlantic coastof the U.S. plus five groups of Mid-Atlantic and New England counties.For simplicity, we will call each of these five groups a “county.” Anyof these 83 “counties” could be the first place a hurricane makeslandfall in the U.S. (or the place another natural peril event occurs).Or none of these counties could be hit by a hurricane. It will bereadily appreciated that the FLO financial activity can be conducted forother numbers of geographical areas, and for other types of naturalperil events where a loss or other significant event occurs, arisingfrom natural forces, i.e. other than man-made, intentional events.

b. First Landfall Options (FLOs) allow parties to hedge or speculate onwhether their selected county will be the first county where a hurricanemakes landfall in the U.S. FLOs are commodity options—the commoditybeing defined in exchange rules to be where a hurricane will makelandfall first. FLOs will be traded on an exchange, a designatedcontract market under the Commodity Exchange Act. Under exchange rulesfor FLOs, a market participant selects one of 84 counties (including the“no hurricane makes landfall” county, called the “null” county) whichthe market participant fears (or believes, or both) will be the U.S.county where a hurricane will first make landfall. That marketparticipant is “long” the county selected and “short” all the othercounties. The market participant pays a premium reflecting this combined“call” on the county selected and “put” on all the other counties. Themarket participant can lose only the amount of the paid premium. If thehurricane makes landfall first in the county selected, the FLO holdersfor that county receive a pro-rata share of the combined proceeds frompremia received and deposited with the exchange for all FLOs purchasedfor all counties in that option series. In other words, all FLOpurchases fund the pay-out to the holders of FLOs for the county wherethe hurricane first makes landfall. As indicated herein, payouts can beadjusted to offer an expected minimum payout, especially for thosegeographical areas in play that suffer verifiable damage.

c. FLO Series. Three options series of FLOs will be offered initially.Based on the historical climatological data. Three FLO option serieswould seem to be adequate because in 90% of the years three hurricanesor less make landfall in the U.S. Each FLO options series would becomprised of options on each of the 83 counties plus the null county.

d. FLO Example. In a first instance, there is only one county where ahurricane makes land fall “first.” Therefore, a FLO buyer for MonroeCounty is “long” Monroe County and “short” all other 83 counties. If thehurricane makes landfall at Monroe County first, the holders of FLOs onMonroe County receive a return on their option. If the hurricane makeslandfall first in any other county, then the holders of FLOs on MonroeCounty receive nothing. Their option expires and can not be exercised.No holder of a FLO can ever lose more than the purchase price (thepremium) for the FLO. If desired, in another example, a natural perilevent can strike two adjacent geographical areas simultaneously, if thenatural peril event is broad enough to do so.

e. FLO Trading Information Market Participants Would Always See. Througha link on the exchange web-site, all market participants will be able toview FLO market data for each of the 84 counties. First, for eachcounty, the market participant would see a percentage number reflectingthe likelihood, based on historical data analyzed algorithmically, thata particular county would be the first county, out of the 84 possiblecounties, where a hurricane would make landfall first in the U.S. orthat no landfall would occur. Second, based on that likelihood for eachcounty, each market participant would see what the FLO for each countythen costs—the premium for the combined call-put option. Third, eachmarket participant would see updated in real-time the number of marketparticipants who have bought a FLO for any county and the proceedsreceived by the exchange to date for all FLOs for all counties. Fourth,on a real-time basis market participants would always be able to see thecurrent value of the FLO for each county. That is, market participantswould know what a FLO holder for Monroe County would receive if noadditional FLOs were bought by market participants and a hurricane madelandfall first in Monroe County. It is anticipated that exchange ruleswill set out the length of the trading day to be 12 hours, from 8 am to8 pm eastern time. Market participants would be able to access theexchange-web site at any time (maintenance may require a short hiatus atsome time during the night) for informational purposes but would be ableto execute FLO purchases only during the 12 hour trading day period. Ifdesired, activities can be conducted for other time durations, such asextending trading to 15 hours, from 8 am to 11 pm, for example.

f. FLO Market Value Variables. The variables that would affect the FLO'smarket value for any county at any time are fully transparent andavailable to all market participants: a) number of FLO purchasers in theselected county; b) number of FLO purchasers in all other counties; andc) the monetary amount paid for all FLOs to date. This information isupdated dynamically on the web-site. No market participant should havean information advantage over any other market participant.

g. FLO Primary Trading: Pre-Tropical Storm Period. Until the firsttropical storm is reported in the Atlantic Ocean, usually July, no datais available on hurricane landfall probability other than historicaldata. Until that first tropical storm is reported, the exchange web-sitewill provide market participants with the price for each FLO county (theFLO premium), including the “null” county, based on an algorithm derivedfrom the historical likelihood that a hurricane will make landfall firstin a particular county. The exchange expects to discount these premiafor the earlier calendar months and then increase the premia as the timefor the expected first reported tropical storm approaches. It would becheaper to purchase a FLO in January, for example, than in April. AssumeMonroe County's historic risk premium is $100, the exchange could decideto reduce that premium for market participants purchasing in January to$50. As one possible alternative, all market participants would (for agiven period of time) be able to buy each FLO for a particular county atthe same price and all would receive the same amount of advance noticeof any discounts granted by the exchange. One exception to this rulemight be for hedgers. The exchange could reduce the FLO premium just forhedgers—those who represent and show that they have property or businessoperations in the FLO county selected (or in contiguous counties). Otherarrangements can be made for the discounting of the premia. These FLOpremium prices would be disclosed to every potential buyer on theexchange's web-site. Buyers also can see on the web-site how the marketforces are driving the potential strike price for each FLO—or what eachFLO holder would receive if the county selected was, at that point intime, the first county where the hurricane made landfall. Thisinformation is updated dynamically on the web-site. No secondary tradingwill be permitted in FLOs during this phase. As a preferred alternative,prices are continuously changing according to the adaptive controlalgorithm described herein.

h. FLO Primary and Secondary Trading: Post-Tropical Storm. Once atropical storm is reported by the National Hurricane Center, theexchange will offer both primary and secondary trading facilities forthe FLOs that are part of the first options series. (When the second andthird tropical storms are reported, the exchange will offer primary andsecondary trading facilities for the FLOs in the second and thirdoptions series, respectively). In primary trading, the exchange willcontinue to publish the prices at which FLOs may be purchased at apremium priced to reflect the historic algorithmic likelihood of thefirst landfall plus the data analyzed from the actual storm trackingsoftware. These primary trading prices will be eligible to be updated atleast four times a day in accordance with the four daily storm trackingreports issued by the National Hurricane Center. In short, the primarytrading price will be an algorithmically-determined combination ofhistory and environment updated dynamically. Secondary trading will bebased solely on offers to sell or offers to buy FLOs received frommarket participants through an electronic auction market trading system.Offers to sell FLOs would have to be made by those that already ownedFLOs, no naked short selling will be allowed. The exchange will providefor clearing of all secondary trades, including the settlement transferof any FLO from the seller to the buyer.

i. Landfall. Exchange rules will establish the criteria, based oncredible third-party observations, for determining when and where ahurricane makes landfall first or a geographical area suffers damagecaused by other types of natural peril events. They are expected to bebased on the publicly-available NHC estimates of the position of thestorm center through time, and on geographic boundaries for the countiesprovided by Census Bureau data bases. Once the hurricane hits, thatevent is over. At some point before the hurricane makes landfall, theexchange rules will have to specify when FLO trading is suspended as thehurricane nears. The exchange does not want market participants incertain regions to be disadvantaged by a storm's imminent arrival.Exchange rules, for example, could specify that primary trading endsonce a hurricane watch is issued for one of the 83 U.S. counties andsecondary trading ends once a hurricane warning is issued for one of the83 U.S. counties. As mentioned elsewhere, public safety and the need tocoordinate various government agencies may require adjustments to theactivities described herein, without departing from the spirit and scopeof the invention.

j. Bounce-back Effect. Sometimes a hurricane will make landfall and thenhead back out to sea before returning to make landfall a second time.For purposes of FLO trading, this occurrence can, in one instance, beconsidered to be two hurricanes, each with a FLO and resulting payout.Exchange rules will specify the criteria for how far out to sea ahurricane must go before it is considered to be a “new” hurricane forpurposes of the FLO product.

k. Null County. Under exchange rules, payouts to holders of the nullcounty FLO occur only at the conclusion of “hurricane season,” generallyconsidered to be December 1. If no hurricane makes landfall in the U.S.,for example, the null county holders for all three FLO series willreceive a pay out from their respective options series premia. If onehurricane made landfall, then the holders of null county FLOs for FLOseries 2 and 3 would receive payouts. Timing of payouts would be set byexchange rules.

l. Other issues. Adding new options series during storm season couldreceive different treatments. For example, a new FLO could be listedwhenever a third storm appears, and a new FLO could be listed when afourth storm appears. Although these FLOs could be funded in a number ofdifferent ways, build up from the early winter sales will be avoidedbecause there will preferably be no winter sales in these FLOs; and theywouldn't be authorized until storm season, in July or August as thelikely earliest time. Either a funding or seed funding source could beprovided or market forces can be relied upon exclusively. In oneexample, it is contemplated that FLOs can be purchased throughintermediary FCMs only, or, alternatively, FLOs can be purchased eitherdirectly by consumers or through the services of an FCM. Marketparticipants can have access either through an exchange web-site orthrough an authorized intermediary (FCM) to execute the purchase of thecommodity option. The FCM can then collect the customer's money and willforward the funds to the exchange or its custodian of the funds.

The concept of a first landfall can be regarded as addressing thecoastal areas that are exposed to hurricane damage. Of course, damageoften continues as the natural peril event travels inland, and provisioncan be made for inland geographical areas as well. Receipts collectedcan be disbursed to a greater number of geographical areas, based on aprior contract or other rules establishing a condition forqualification, as where a natural peril event impacts inlandgeographical areas. Such impact could be qualified based upon anofficial track, or a minimum amount of damage or damage claims, forexample. Such damage could also be caused by a tornado, an earthquake,or other type of natural peril event.

20.2 Trading Utilizing a Severity or Property Damage Scale

A second example of a derivative trading financial example alsocontemplates financial activity concerning property owners' exposure orrisk of exposure to damage related to hurricanes and other naturallyoccurring events. It has long been recognized that a hurricane's ortornado's potential to inflict damage to property can be estimated fromvectored wind speeds, such as translational and rotational wind speeds,and other observed characteristics such as the storm's radius or othersize-indicating data. A scale or index of measurement is derived from aevents of observed facts concerning hurricane characteristics. Theobserved facts are preferably provided by a responsible third party,such as a government agency or weather information specialist service ora damage prediction service, for example. Examples of observed factsinclude temperature, humidity, wind speed, and wind area relating to astorm. Observed facts also include storm intensity forecasts provided bythe National Hurricane Center, for example.

Values on the scale provide an indication or measurement of the risk ofproperty damage as a result of the exposure to an historical orforecasted hurricane (or other like natural peril event). In oneinstance, the values are calculated or computed numbers and the scale ofvalues may be in the form of an index, table or other data construct.The indication or measurement of the risk of property damage can, forexample, relate solely to the likelihood of landfall, or could relate tothe prospect of landfall with or without consideration of stormintensity and with our without consideration of property values exposedto the storm. In another instance, indication of damage is based uponactual or estimated insurance claims. Claims for certain types of damage(hurricane or flood or earthquake) could be excluded, if desired.

In one variation of financial activity according to principles of thepresent invention, derivative products such as futures (forwardcontracts), and options (option contracts) are traded multilaterally(one or many trading with many) or bilaterally via an exchange or otherdealer or privately, i.e. directly, between a buyer and seller. As afurther alternative, a manager of a financial activity may allow privatecontracts and other dealings to be later cleared through an exchange'sclearing system. Preferably, the various contracts are settled against ascale, expressed, for example, as an index, a table, or other dataconstruct, that is calculated with reference to observed facts relatingto natural peril events, such as those mentioned elsewhere, herein. Thescale could relate, for example, to the likelihood of first landfall, orthe potential of the naturally occurring event to cause damage toproperty exposed to the naturally occurring event. The scale could beemployed as an index, used in the manner of conventional futurestrading, with financial activities modeled after conventional futurestrading (i.e. forward contracts and options contracts) in indexes. Theindexes could, for example, be allowed to vary in value as the naturalperil event matures, with trading being settled according to indexvalues on separate days

The scale can be expressed in terms of continuous or discontinuousnumerical series. Virtually any basis for the scale(s) can be used. Forexample, scales or indexes measuring expected property damage can bebased on meteorological data such as velocity (e.g. high wind speeds)and overall size such as radius data for storm (hurricane) winds. Suchdata can include, for example, a single radius of hurricane force winds,or arrays of hurricane force winds in a more complex analysis. Otherbases for the scales could chosen from data relevant to insuranceinterests, such as property damage claims as well as establishedinsurance and reinsurance indicators of damage, damage risk as well asclaims presented for settlement according to accepted insurancepractices. If desired, the financial activity can employ more than onescale. For example, different scales could be used at different stagesof a hurricane's progression, such as the storms development at sea,first landfall (usually, for a specified region). Multiple scales couldbe employed, for example, where a number of indexes are put in play forsuccessive hurricane landfalls in a given season. Each index could beseparate from the others, and allowed to fluctuate according to ongoingmarket valuations and activities. For example, the indexes couldfluctuate in value according to market performance of the financialactivity, or alternatively, according to technical information such aschanging forecasts by the NHC or other independent third party

The separate indexes could be settled separately, for example, shortlyafter the hurricane makes landfall. Different scales could also relateto the overall cumulative effect of natural peril events over a periodof time, such as a hurricane season, for one or more regions ofinterest. Further possibilities include scales relating to a largest ormost powerful or most sustained storm in a particular area, as definedby the activity. If desired, a cumulative long-term index based upon twoor more shorter term indexes, can be traded. Alternatively, a cumulativeindex can be based upon individual indexes for two or more geographicregions. If desired, indexes based upon permutations or combinations ofdifferent types of indexes can be used, including indexes for differentgeographic regions and indexes for different time durations.

Contracts for the futures and options on futures, and other financialactivity products are preferably created for defined geographicalregions exposed to hurricane and other naturally occurring events. Ifdesired, where hurricanes are the naturally occurring event of interest,the scale(s) employed could relate to the hurricane's first landfall, arange of geographical regions over which all hurricanes for a seasonmake landfall, or geographic regions suffering exposure to a minimumstorm force. First landfall could be on a geographicregion-by-geographic region basis, or on the basis of all definedgeographic regions taken together, or in multiple-region subdivisions orother defined manner. If desired, derivative products or other financialactivity could be defined for natural pre-hurricane events. For example,derivative products can be defined when a storm activity reaches atropical depression stage as determined by the NHC or other authorities.Further, such products can be defined for different geographical areasdefined by the activity, such as storms occurring in different regionsof the sea, such a more northerly Atlantic regions and more southerlygulf coast regions. Such activities could be tied to market interestsrelating to more southerly gulf coast exposures as compared to morenortherly Atlantic coast exposures. Trading for such products cancontinue, for example until landfall or until the storm expires at seawithout making landfall. As a further alternative, products related tonatural peril events originating in one geographical region can beexpired or converted into different products when the underlying naturalperil event leaves one geographical region and enters another.

In one example, the financial products including derivative contractscan be delivered or otherwise settled in a variety of ways. For example,they can be settled with reference to a final price determined accordingto a scale, or according to market activity or according to one or morealgorithms or any permutation or combination of these. Further,settlement can be made in the context of a pari-mutuel activity, or afixed settlement method, or a combination of these. If desired, thefinancial activity can allow, or regulate or facilitate secondarytrading, where financial products are traded between participants beforeor after settlement. In one example, futures and options on futuresbased on first landfall of a hurricane with respect to one or moredefined regions, are traded during a hurricane season, prior to firstlandfall. Market forces can control prices in this secondary trading, ifdesired, or prices can be regulated by the financial activity, in somemanner. Settlement is made after first landfall, either at a fixed priceor a price determined by the financial activity, for example, accordingto principles set forth herein.

In one variation of financial activity according to principles of thepresent invention, financial products such as futures contracts, optionscontracts and options on futures contracts, are cleared through afutures exchange. The contracts may be created for defined geographicalregions exposed to hurricanes and other naturally occurring activity,and are preferably settled against one or more embodiments of a scale ofmeasurement of the type described above, or herein, derived from aevents of observed facts concerning characteristics of natural perilevents, especially characteristics related to a potential of a naturallyoccurring event to inflict property damage. Values on the scale may beexpressed in the form of an index, a table, or other data construct asmay be desired. The contracts may set a lower limit on the amount ofdamage contemplated. The contracts may be limited to future potentialdamage exposure, or may be implemented for actual sustained damage,preferably damage estimates provided by a responsible external thirdparty, such as a government entity or insurance or reinsurance service.If desired, the scale may be based in whole or in part on informationprovided by such government entity or insurance or reinsurance service.

In general, contracts considered herein could be defined in terms ofactual or estimated future damage, with or without a set minimum amountof damage, or the likelihood that a defined amount of damage is reportedfor a particular storm, a range of dates or an entire season. Ifdesired, the contracts could be defined on a storm-by-storm basis, orcould be limited to the storm having the greatest impact in the chosencategory (damage caused, size of storm, strength of the storm, measuredin some predetermined manner, size of coastal area or land areaaffected). If desired, contracts could also include offshore propertiessuch docks, or harbors, or ships that are moored, tethered or otherwisepresent within a defined proximity to land.

If desired, multiple scales can be provided for each geographic region,or different time divisions, such as different date ranges, and could beprovided for different natural peril events, in which each firstlandfall of a season terminates the current part of the financialactivity, with a new, subsequent part being started. In each of thevariations herein, contracts could also be offered for a “null” event inwhich no landfall is reported for a given season or other defined timeperiod.

20.3 Sarasota County Example

As a further variation, one financial activity according to principlesof the present invention contemplates trading of contracts utilizing afutures index. As will be seen herein, the contracts are written for afirst landfall criteria, although different criteria based on othertypes of natural peril events could be employed, as well. This variationassumes interest in a particular geographical region, referred to hereinas Sarasota County. It is assumed in this variation, that, although thestorm is still some days away from occurring (e.g. away from landfall),that it is becoming more likely that the storm may be heading forSarasota County.

An index is provided as a basis for trading geographical regions, andfutures contracts are offered for trade. The Sarasota County Index risesas it becomes more likely Sarasota will be First Landfall (or otherwisesubjected to damage), and falls as it becomes less likely. The futurescontract on that Index are based on the proposition that, when the firststorm makes landfall, all futures contracts expire and buyers or sellersmust pay off based on what the last index price was before Landfall. Thevariation focuses on two participants, a first participant that thinksSarasota will be hit first, and a second participant that doesn't. Thefirst participant buys a Sarasota Index futures contract at $100 onMonday, and accordingly is long on Sarasota County The secondparticipant sells a Sarasota Index futures contract at $100 on Monday,and accordingly is short on Sarasota County. Both participants paymargins as required by exchange rules and by contracts with theparticipant's “clearing firms.” For discussion purposes, eachparticipant pays $5 in margin to their clearing firm.

By Thursday, it looks like Sarasota will be hit over the weekend. TheSarasota Index rises. The price of the Sarasota futures contract alsowill rise. It is assumed that, based on bids and offers from all marketparticipants, the price for the futures contract on Thursday is $125. Ifthe first participant decides to settle for this profit, the broker isinstructed to liquidate the first participant's future position throughan offsetting trade. The broker for the first participant then goes intothe market and sells a second Sarasota contract at $125 on behalf of thefirst participant. The resulting sell contract cancels out or offsetsthe first participant's buy contract. In the sale, the buyer is assumedto be a third participant. The first participant receives a profit of$25 (along with return of the margin paid).

On Friday, the storm turns sharply and heads towards Texas. The SarasotaIndex plummets. The price of the Sarasota future is assumed to drop to$10. Assuming the first participant wants to maintain a position onSarasota County, and the first participant's broker finds a participantwilling to sell at $10, the resulting buy order at $10 cancels the sellorder at $100 and leaves the first participant with a profit of $90.

20.4 Additional Considerations

In one example, once an investment is made for a particular geographicalarea, the next investment to be made for that same geographical area isset at a higher purchase price. For example, the rules of the financialactivity can provide that the second investment will undergo a flat ratepricing increase, such as $0.25 per financial investment unit. Ifdesired, pricing increases can be assigned in steps according to stepincreases in the volume (either dollar amount or number of financialinvestment units traded) of trading. If desired, other types ofincreases can be employed, including linear and nonlinear mathematicaltreatments of purchase price. The prices for subsequent investments in aparticular geographical area may either continue to increase or willplateau at a constant price until a set point minimum number offinancial investment units is reached. In this example, preferably, theset point is chosen to reflect a pool of money of substantial size,which would justify a substantial step in price increase. If desired,several steps of price increases can be employed and related to similaror different sized blocks of financial investment units. If, forexample, a particular geographical area suffers a lull in trading, thepurchase price can be reduced, based on the time value since the lasttrading activity. Once activity resumes for a particular geographicalarea, price increases can go into effect. Each geographical area willhave different initial prices and could have different step sizes orblocks, as financial investment units are traded, if desired. Thus, inthis example, financial investment unit price varies according to marketactivity (e.g. a stepwise or a smoothly varying increase as tradingactivity increases). Optionally, in this example financial investmentunit price can also vary according to time factors, such as factorscentered around a falloff (or optionally a rise) in trading activity.

It is generally preferred that pricing follows a number of principles.For example, it is generally preferred that trading begins with a setminimum price which could be different for each geographical area. Inone instance, it is generally preferred that the minimum price dependson market activity. In a second instance, it depends on theclimatological probability, (as currently estimated), of the particulargeographical area being hit, and, as trading progresses, how manyfinancial investment units have been sold for a particular geographicalarea, and also the total amount of money in the pot. If desired, each ofthese principles can be applied in different amounts, (e.g. according todifferent increments, different rates or other mathematical treatment).It is generally preferred that the minimum price increases as theprobability for a hit in the geographical area increases, and theminimum price should also go up as the total dollars in the pool ofmoney goes up. In one instance, it is generally preferred that theminimum price goes down as the number of financial investment unitspotentially splitting the pool of money (i.e. for a particulargeographic area) goes up. Other considerations regarding pricing arementioned herein.

20.5 Auction-Based Financial Activities

As another example, in a maturing market, the ongoing financial activitycan be alternated (less preferably, replaced) with an “auction” form offinancial activity. In one instance, the auction form of financialactivity, once initiated, is scheduled to occur at different intervals.For example, multiple auctions would preferably occur periodicallythroughout the season, in one instance, with the time spacing betweenauctions being sparser (e.g. weekly or monthly) early in the process,and more frequent (e.g. multiple auctions per day) when tropicalcyclones are in existence and especially when hurricanes are threateningimminently. this example, the event price for an auction is preferablymade proportional to the probability that a particular geographic areawill be hit, multiplied by the funds available, and that product dividedby the number of financial investment units outstanding for thatparticular geographical area. If desired, the proportionality constantapplied could be unity or some number smaller or greater than one.

Variation 1: Assuming a first auction has concluded at a time inJanuary, before the conventional hurricane or other natural peril eventseason, the “stage 1” probability, calculated as set forth herein, is0.02. Assuming the previous financial activity sold 5000 financialinvestment units for the geographical area of interest, and thatcollectively, all the auctions took in $25 million overall. The minimumunit price would then be (0.02)×($25 million)/5000=$100, and therefore,nobody could bid less than a $100 minimum price. If there were moretakers than financial investment units, some or all of the bidders at$100 would be out of luck.

Variation 2: Assuming a time in July, when there is a tropical storm orother natural peril event in existence, in a location that is favorablefor hitting the geographical area (e.g. county of interest. The “stage2” probability for that area is calculated as set forth herein to be0.10. There are now 100,000 financial investment units that have beenbought for the geographical area of interest, and the total pool ofmoney is $2 billion. Minimum bid price would be (0.10)×($2billion)/100,000=$2000.

Variation 3: That tropical storm or other natural peril eventcontemplated in variation 2 has dissipated and now poses no threat tothe geographical area of interest. Also, there are currently no otherstorms or other natural peril events that are threatening, and theprobability drops back to 0.02. There's now $2.1 billion in the pot, and105,000 financial investment units have been bought for the geographicalarea of interest. The minimum bid price has gone down because theprobability has gone down: (0.02)×($2.1 billion)/105,000=$400.

In each of the variations above, there would preferably be separateauctions for each geographical area in play (with one or more auctionsfor each geographical area), with the degree of market participantinterest determining whether and/or how much the price would rise abovethe levels indicated. Also, it is generally preferred that the number ofunits on offer at any one time would be limited to a fixed number, itbeing expected that, if the total offer is unlimited, there would be noincentive to bid higher than the price.

If desired, special treatment could be given later in a financialactivity season when it becomes clear that certain geographical areasare unlikely to ever be perceived as generating substantial marketparticipant interest (e.g. inland geographical areas that historicallyhave not, or only very rarely, been hit.

20.6 Additional Examples

In another example, trading contracts corresponding to differentgeographical sections, have contract prices that are preferably set bymarket forces, that is, by the participants, rising and falling based ontheir degree of confidence of where landfall will occur. The value ofthese contracts will rise and fall based on market acceptance of aforecasted storm occurrence.

In a further example, derivatives trading is based entirely on“market-based” pricing. Here, “market-based” pricing refers to pricingthat is based on recent or cumulative market activity. If desired, themarket-based pricing may or may not be driven by algorithms, or onlypartially driven by algorithms, as may be desired.

In another example, futures contracts, options contracts, options onfutures contracts and other forms of derivative products may pay outupon occurrence of a specified condition (e.g. first landfall of ahurricane), preferably on a non-pari-mutuel basis, are offered for saleto prospective purchasers. Most of the contracts represent differentcoastal areas subject to first strike by a hurricane, typically,portions of the Atlantic and the Gulf seacoast. If desired, contractsfor non-landfall conditions can be added to the offering, such as astorm at sea entering a defined zone. Prices of the futures contractswill vary during a storm season, with prices for particular contractsrising according to market activity as a storm approaches the territory(or other condition) defined in the futures contract.

As can be seen from the above, the present invention contemplates anumber of features concerning derivative products, including financialactivities directed to catastrophic events, especially non-human created(e.g. “natural”) events (landfall for a hurricane, location for aearthquake or tornado, etc.), a prediction of the location and/or thepotential for disaster of the future event. Settlement is determinedagainst information provided by an independent, responsible third party(that provides, for example, a determination of landfall, location ordamage amount). Settlement may be of a pari-mutuel type, market-driventype, or a mixture of market forces and pari-mutuel types. The financialactivities may contemplate scales or indexes concerning the subjectmatter involved, such as a third party estimate of the degree of damagesustained (such as the amount of insurance claims asserted or an indexbased upon such data), or the severity of the event (such as wind speedsor other damage potential of a hurricane). Derivative activities may beconducted between pairs of participants, or between unequal numbers ofparticipants or between one or more participants and a financialinstitution. Derivative activities may involve algorithms, such asalgorithms for setting pricing of some portion or all of an activity, ormay react solely r partially to supply and demand and other marketforces. The above features may be entirely or partially aggregated invirtually any permutation or combination, or may be left out altogetheror may be replaced by other features contemplated herein.

In other aspects, the present invention contemplates futures contracts,options contracts, options on futures contracts and other forms ofderivative products, traded multilaterally or bilaterally, where thecontract's or product's value increases or decreases based upon ameasure or index of estimated or actual: property damage or likelihoodof property damage, whether caused by a weather event or other form ofnatural peril, in a specified region or geographic locations. Thecontract's or product's value may increase or decrease based upon ameasure or index concerning where a weather or other natural peril eventwill occur (e.g. make landfall first) in a region or geographical area.The contract's or product's value may also increase or decrease basedupon what the seasonal impact of weather or other natural peril eventswill be in a specified region or geographic area and may also be basedupon what the maximum or largest weather or other natural peril eventwill be in a specified region or geographical area. Again, permutationsand combinations of these financial activities are possible.

XXI. “Hurricane Pools” Financial Activity

21.1 Summary

The following description is directed to examples of financialactivities that are referred to as “Hurricane Pools” or “Storm Pools.”As mentioned, different types of financial activity models may beemployed. The following example is given in terms of a financialactivity model of the type comprising a game of skill, although many ofthe principles set forth herein are applicable to other financialactivity models, as well, such as the preferred First Landfall Marketmentioned herein. The financial activity contemplated herein isgenerally referred to as “Hurricane Pools”, emphasizing the investmentnature of the financial activity as a hedge against unforeseen loss. Inthe example given, the Hurricane Pools are games of skill that focus ona particular type of natural peril event, namely hurricanes makinglandfall in the United States. That is, in this example, only tropicalcyclones having a strength meeting the minimum criteria to be termed“hurricanes” are considered. Those skilled in the art will readilyappreciate that other types of natural peril events can be treated aswell.

It is recognized that a tropical cyclone originates at sea, and grows inintensity over time, before making landfall. It is possible that theintensity of the tropical cyclone may rise and fall. The criteria chosenhere is that the tropical cyclone has a minimum category one hurricaneintensity at the time of landfall. The Hurricane Pools are structured toallow market participants to use them in a way that can augmenthurricane insurance, while simultaneously providing income toparticipating states to help defray costs associated with disastermanagement. Homeowners and business insurance policies typically containdeductible provisions ranging from 2% to 15% of a home's value. Inaddition, these same policies do not provide any coverage for theoutside areas of a home or business, such as landscaping, outsidelighting, docks, fencing and the like. Often, property owners do nothave sufficient flood insurance and have other omissions or insufficientcoverage which result in catastrophic financial losses in even thelowest rated hurricanes. Also, losses are suffered when rates are lostand also, where temporary housing is needed.

Because financial activities can be carried out according to differentfinancial models. The Hurricane Pools, for example can be distinguishedfrom insurance instruments, with payouts for qualifying investments notbeing tied to actual property losses, and thus being dispensed morequickly. Payouts here do not involve inspection by adjusters, andtherefore can be made much more promptly—(e.g. within a few weeks). Thispromptness and flexibility can be achieved because the Hurricane Poolsoperate in a way that emphasis is placed on payouts that primarilydepend upon apportioning Hurricane Pools that have been invested into aparticular Hurricane Pool, and on the amounts and timing of HurricanePool entries for counties that experience a hurricane strike.

As mentioned above, financial activities may be constructed aroundmodels that cover the natural peril event activity either on an event byevent basis, or on a “seasonal” basis for events occurring during aredefined span of time. In the example given above, choice was made tooperate on the “seasonal” basis. Accordingly, the number of HurricanePools in which market participants may participate can vary from year toyear, depending on the number of officially declared eligible events. Inthis instance, the Hurricane Pools may be regarded as different eventsoccurring in a season. Initially, two or three Hurricane Pools may beopened to investment, beginning on 1 January. Each of these initialHurricane Pools relate to one of the first two or three tropicalcyclones passing over a portion of the U.S. with at leasthurricane-force winds during the upcoming calendar year, as determined,in one instance, from the tropical cyclone Advisories issued by theNational Hurricane Center (NHC). The overwhelming majority of theseevents occur during the span of time, popularly referred to as the“hurricane season” (1 June through 30 November). In active hurricaneseasons, additional Hurricane Pools may be opened as the hurricaneseason progresses.

Each investment in a Hurricane Pool is preferably made in the form offinancial investment units or the like, purchased for one or more of thegeographical regions, such as counties (or county equivalents, forexample Louisiana parishes) that are plausible hurricane targets. Whenthese financial investment units have been purchased for counties thatlater receive a hurricane strike, they qualify their owners to receivepayouts from that Hurricane Pool. All of the investments in a givenHurricane Pool, less items which may be defined in a given instance(e.g. portions designated for participating state governments and feesfor Hurricane Pool management) are paid to market participants in thequalifying county or counties. The formula determining these payoutsaccounts both for the chances of a hurricane striking the qualifyingcounty(s), as assessed at the time a Hurricane Pool investment is made,and a reward for earlier investments into that Hurricane Pool.

21.2 Structure of Hurricane Pool Investments

A. Hurricane Pool Financial Investment Units

Entries in the Hurricane Pools are made for individual counties, in theform of “financial investment units” in one of the specific HurricanePools available for investment. The first Hurricane Pool relates to thefirst tropical cyclone to make landfall as a hurricane over the U.S.,including Puerto Rico and the U.S. Virgin Islands; the second HurricanePool relates to the second hurricane striking the same area; and so on.For example, suppose the first U.S. hurricane landfall in a hypotheticalyear is at Galveston County, Texas, and the second is at Miami-DadeCounty, Florida. Financial investment units purchased in Hurricane Pool#1 for Galveston would qualify their owners for a portion in all of themoney invested in that Hurricane Pool, but market participants inHurricane Pool #1 for Miami-Dade would receive nothing from thatHurricane Pool. On the other hand, market participants in Hurricane Pool#2 for Miami-Dade would be entitled to a portion of all of the moneyinvested in that Hurricane Pool, whereas Galveston market participantsin Hurricane Pool #2 would receive nothing from that Hurricane Pool.

The primary means of making Hurricane Pools investments will, in oneexample, be by credit card, through a Hurricane Pools website. Oneexample of a web site is given in FIGS. 13-18, which shows a web siteimplementing a Hurricane Pool financial activity. FIG. 13 shows a website screen which serves either as a welcome page or one of the firstpages that a participant will encounter upon acquiring the web site.Included in the screen depicted in FIG. 13 is an indication 600 of thecurrent open Hurricane Pools and a brief summary 602 of current tropicalcyclone activity.

In one instance, in addition to providing web site access, individualswho do not have internet access will also be able to participate in theHurricane Pools by using touch screen displays and other examples ofgraphical user interfaces, located at convenience stores, gas stationsand the like. Individuals will make selections by touching aninteractive screen, for example, and pay for their investment by swipinga credit card or providing a cash payment to the retail establishment.Preferably, the display will automatically generate a printed receipt(including identification number) for both credit card and cashpurchases. A social security number and perhaps a biometric device suchas a fingerprint scan may be required to participate in the HurricanePool.

FIGS. 16-18 show financial investment unit purchases for the Storm Poolweb site implementation of a Hurricane Pool. In FIGS. 16 a and 16 b-16 ctwo screens are shown for purchasing financial investment units and inFIG. 17 a confirmation is given for financial investment unitspurchased. In FIG. 18, a summary or “portfolio” of all transactions fora participant is shown.

Once an investment in a Hurricane Pool is made, the investmentpreferably cannot be reversed as these would affect the value of thefinancial investment units purchased by others. In one instance, aninvestment is not considered to have been accepted until the marketparticipant's credit card company credits that investment in theHurricane Pool. If the credit card company later reverses that paymentto the Hurricane Pool, the value of the financial investment unitspurchased in the Hurricane Pool will be preferably set to zero.

B. Financial Investment Unit Prices

The price of a Hurricane Pool financial investment unit for a particularcounty is, in one instance, determined by a mathematical formulainvolving both the probability of that county being hit, and optionalprice discounts for early investments. Therefore, financial investmentunit prices in Hurricane Pools will be different at different times anddifferent for different counties at a particular time.

The price incentives for early investment may be substantial, and aredesigned to encourage investments before any tropical events are inexistence, and indeed well before the beginning of hurricane season. Itis anticipated that much of the early investment activity will come frommarket participants who may want to use the Hurricane Pools tosupplement conventional insurance, or from insurers using the HurricanePools as a reinsurance vehicle. One purpose of financially penalizinglater investments is to obtain the greatest amount of money in theHurricane Pool as possible, by encouraging early investing anddiscouraging procrastination.

Financial investment unit prices are preferably higher when hurricanestrike probabilities are higher, and lower when hurricane strikeprobabilities are lower. Specific details of the pricing formula willpreferably be made available on the Hurricane Pools website, for thosewho may be interested. Factors influencing the pricing probabilitiesinclude the following:

1. The location of the county for which the investment is being made.For example, counties in south Florida are historically more likely tobe hit than are counties in Massachusetts, on average, and so financialinvestment unit prices for Florida counties will usually be higher.

2. The size of the county. Larger counties present bigger targets, andso financial investment units in larger counties will generally be morecostly.

3. Which of the Hurricane Pools an investment is made in. For example,financial investment units in Hurricane Pool #2 cost less than financialinvestment units in Hurricane Pool #1, other factors being equal,because it is more likely for one U.S hurricane to occur in a given yearthan for two such events to occur.

4. The location(s) and strength(s) of tropical cyclones in the AtlanticOcean, Caribbean Sea, and/or Gulf of Mexico, that potentially threatenthe United States. Such storms in some locations are more likely toaffect particular counties as hurricanes, based on a historicalclimatological analysis of more than a century of hurricane data, andfinancial investment unit prices for such counties will increaseaccordingly. Once a tropical depression is announced by the NationalHurricane Center, and continuing through the intensification of thestorm, financial investment unit prices continue to increase each time astorm is upgraded in strength, because threats to land are larger forstronger storms. This aspect of the financial investment unit pricing isintended to prevent persons who now have knowledge of a currentlyexisting storm from being unfairly rewarded by having this information,relative to early Hurricane Pool market participants.

The above pricing considerations may, in one example, be automaticallydetermined if desired, by recognizing that market activity presumablyreflects these considerations, as viewed by market participants.Accordingly, market activity can be relied upon as the sole determinantof these foregoing considerations, with the assumption that theseconsiderations are incorporated into the market decisions made byparticipants. After an initial period of price setting, to originatemarket activity, the number of options or other units shares availablefor purchase indicate the aggregate analysis by the investing community.

C. Opening and Closing Hurricane Pools

The number of Hurricane Pools that may be opened to investment is at thediscretion of the Hurricane Pools management or provider. It isanticipated that two or three Hurricane Pools will be opened on January1st of each year. These Hurricane Pools pertain to possible U.S.hurricane landfalls during that calendar year, whether or not they occurduring the generally recognized “hurricane season” (June 1st throughNovember 30th). At the discretion of the Hurricane Pools Administrators,additional Hurricane Pools may be opened before the beginning of the“hurricane season,” for example if an unusually large number ofhurricanes may be forecast to occur in that year. Similarly, additionalHurricane Pools may be opened to investment during the hurricane season,particularly if previously open Hurricane Pools have all been closed byhurricane landfalls (preferably, measured by the eye of the hurricanehitting land), or by imminent possible hurricane landfalls.

Hurricane Pools will preferably be closed to further investment when thestorm to which they pertain is sufficiently close to landfall, accordingto the current NHC Advisories for that storm. Preferably, persons shouldleave home and not participate in the Hurricane Pools activity, whentold to evacuate. The exact trigger for Hurricane Pool closingspreferably strikes a balance between public safety (not discouragingprudent evacuation by remaining open too long) and broad participation(not cutting off investments before a given storm is an imminentthreat). A possible compromise could be to trigger a Hurricane Poolclosure when its tropical cyclone is both at hurricane strength and hasgenerated NHC hurricane warnings for one or more of the counties forwhich Hurricane Pool investments may be made. In addition, forfast-moving and/or rapidly developing hurricanes, Hurricane Pools wouldbe closed when the operational estimate of its track, as published inthe relevant NHC Advisory, has traversed at least one of the countiesfor which Hurricane Pool investments may be made.

Because Puerto Rico and the U.S. Virgin Islands are relatively far fromthe U.S. mainland, a Hurricane Pool may be closed for these twoterritories without it necessarily being closed for the rest of the U.S.In such cases, financial investment units for counties in theconterminous U.S. that are subsequently traversed by the same storm athurricane strength will also qualify to receive payouts from thatHurricane Pool. However, a Hurricane Pool that is closed because ofstorm proximity to or landfall on the U.S. mainland will preferably alsoclose for Puerto Rico and the U.S. Virgin Islands.

21.3 Hurricane Pool Payouts

A. Disbursements to Qualifying Financial Investment Units

Preferably, all of the money invested in a given Hurricane Pool, lessfixed percentages for participating state governments (to help supportemergency management efforts) and for Hurricane Pool management, isdivided equally among Qualifying Financial investment units. AQualifying Financial investment unit is a financial investment unitpurchased for a county that is subsequently hit by the hurricane towhich that Hurricane Pool pertains. Therefore early market participants,whose Qualifying Financial investment units were purchased relativelycheaply, will realize larger returns on their investments than willlater market participants, for whom the financial investment unit priceswere higher.

To the extent possible, disbursements front Hurricane Pools to holdersof Qualifying Financial investment units will preferably be made byposting the amounts to the credit card account from which the investmentwas originally made. This mechanism has the advantage that thedisbursements will be available very quickly, to people who may needthese financial resources for rebuilding or other hurricane-relatedexpenses. For example, individuals who may be displaced by extendedevacuation from their homes may have especially acute needs for thesepayouts. These problems are magnified by loss of jobs. It may benecessary to withhold portions of large disbursements on behalf of theIRS.

B. Determination of Qualifying Financial Investment Units

For the purposes of determining Qualifying Financial investment units inHurricane Pools, counties are considered to have been “hit” in oneinstance if the track of the center of the hurricane as determined fromthe NHC Forecast, Public, or Special Advisories for that storm (or, e.g.within, +/−20 nautical miles) passes through some portion of thatcounty. The basic information in the NHC Advisories that is used todetermine Qualifying Financial investment units in this exemplaryinstance are the storm positions (latitude and longitude) and strengths(maximum sustained winds), that are reported to have occurred atparticular times. These location points will preferably be connected bystraight-line segments (or, optionally curves calculated from a formulain operational use at NHC or otherwise made public and preferablymentioned by reference on the Hurricane Pool website). Financialinvestment units in counties traversed by a track so calculated, betweentwo consecutive positions at which the storm was at hurricane strength(maximum sustained wind of 64 kt, or 74 mph), will, in this instance, beQualifying Financial investment units. For this purpose, thegeographical extent of counties will be defined for example by theCartographic Boundary Files of the U.S. Census Bureau, that arepublished at http://www.census.gov/geo/www/cob/co2000.html. Thesedefinitions could also be modified to allow financial investment unitsin counties affected by stronger hurricanes (Category 2+ hurricanes, asindicated by maximum sustained winds reported in the NHC Advisories) toreturn a larger financial investment unit of Hurricane Pool assets totheir market participants.

Preferably, it should be pointed out to Market participants in HurricanePools that these rules for determining Qualifying Financial investmentunits have been somewhat idealized, relative to real-world hurricanebehavior, in the interest of having a promptly available, clear,explicit and automatic way of disbursing Hurricane Pool assets toQualifying Financial investment units. In particular, a few pointsshould be noted. The first point is that: there will often be countiesexperiencing hurricane-force winds and/or other hurricane impacts thatnevertheless do not qualify as having been “hit” according to thedefinition used by the Hurricane Pools. This will be the case especiallyfor the larger and more powerful storms. Hurricane Pool marketparticipants whose intention is to, in effect, supplement theirinsurance coverage are therefore encouraged to invest in surroundingcounties also. To encourage market participants to protect themselves,the Hurricane Pools site will automatically flash or change color forseveral counties which border the county initially selected and urge ourmarket participants to spread their investment to include surrounding(collar) counties. In this manner, the market participant will, in thisinstance, have greater opportunity to collect from the Hurricane Poolsactively if damage occurs, but the eye of the hurricane does not entertheir county.

The second point is that, because qualifying counties are determined onthe basis of storm positions only at particular, and possibly irregulartimes, small discrepancies between the calculated track (used todetermine Qualifying Financial investment units) and the actual track(as determined some months later in the official NHC Tropical CycloneReport for that storm, or that might be evident at the time of the stormfrom a events of weather-radar images, for example) can and will occur.Again, it may be advisable for some individuals to invest in nearbycounties, in addition to the county(s) in which they have the mostinterest.

As a third point, the U.S. Census Bureau data files are onlyapproximations to the true geographical outlines of many counties. Theyconsist of a collection of line segments, and so will not accuratelyfollow curving county boundaries. In addition, portions of some counties(particularly relatively small islands) are not included in the CensusBureau's Cartographic Boundary Files. For example, the Dry Tortugas arenot included in the Cartographic Boundary File for Florida, so that ahurricane passing over this portion of Monroe County, Florida, would notby itself constitute a “hit” on Monroe County for the purpose ofdetermining Qualifying Financial investment units.

In cases where there may be multiple tropical cyclones in existence atthe same time, an explicit rule deciding which storm pertains to whichHurricane Pool is needed, if more than one of them eventually affectsthe U.S. as a hurricane. Priority can be determined according to thetime of landfall on at least one of the counties for which HurricanePools investments may be made. For example, if the hypotheticalhurricane “Bob” makes U.S. landfall ahead of hypothetical hurricane“Alice,” “Bob” would be assigned to Hurricane Pool #1, and “Alice” wouldbe assigned to Hurricane Pool #2.

It is anticipated that payouts to Qualifying Financial investment unitswill be made within two weeks of the final NHC Advisory for the storm inquestion. In unusual cases, such as for storms that may have thepotential to reintensify and affect the U.S. again, Hurricane Poolpayouts may be delayed beyond two weeks at the sole discretion of theHurricane Pools Administrators. In all cases, Hurricane Poolsdisbursements will preferably be made on the basis of the best and mostrecent information available from the National Hurricane Center at thattime about the storm in question, and will not be subject to revision inthe event of subsequent updates to that information.

21.4 The Hurricane Pools Website

With reference to FIGS. 13-18, the Hurricane Pool's activity, in oneinstance, will be administered through a website that calculatesfinancial investment unit prices (or prices of other financialinvestment units) automatically, according to an algorithm that tracksmarket activity. As an initial stage, or concurrently in later stages,information from NHC Advisories that are updated at least 4 times dailycan be considered as a pricing factor when one or more Atlantic tropicalcyclones are present, and receives payments from market participants'credit card accounts, or other financial equivalent, or otheralternatives to payment, such as transactions to an market participant'sbrokerage account. Preferably, first-time entrants will need toregister. Password protection will preferably be employed forcredit-card accounts associated with each registration, to facilitateany eventual payouts that may need to be made to that account. SocialSecurity information will preferably be required as part of theregistration, in order for state and national government agencies totrack tax liability on any payouts. Other arrangements could be made forthose who access the website through their brokerage service or otherthird party service.

Current financial investment unit prices for all available counties orother geographic regions can be displayed both graphically and intabular form. With reference to the schematic screen depictions of FIGS.14 and 15, clickable maps (coastal area and individual state) are madeavailable, with financial investment unit prices indicated approximatelywith a color code. Preferably, state-by-state pull down menus (notshown) are also provided. Participants are to be given the option ofspecifying their entries either in terms of financial investment unitsbought, or dollars to be entered, for each county selected. Choices madeby a participant are shown in shaded form in FIGS. 14 and 15.

The sums entered to date in each Hurricane Pool (and available forsubsequent payout) will be posted on the Hurricane Pools website andcontinuously updated. For Hurricane Pools that are currently open forinvestment, it will be possible for site visitors to determine potentialpayouts for a financial investment unit in any county, under variousassumptions about the track of the hurricane to which that HurricanePool pertains. These “what if” calculations can be made available forany historical hurricanes that have crossed a county in question, or forany hypothetical hurricane track that a website visitor might beinterested in.

For Hurricane Pools that have been closed to further investment, theactual hurricane track and Hurricane Pool payouts per financialinvestment unit will preferably be listed, together with a variety ofofficial NHC information about that storm. Until a given tropicalcyclone has fully dissipated, and so has no chance of reintensifying andsubsequently affecting a portion of the U.S., these estimates will besubject to revision.

Because financial investment unit prices will be updated (preferably atleast four times daily according to the most current Advisoryinformation from NHC), it will be necessary for the site to beunavailable for accepting investments for short, pre-scheduled, periodsof time (e.g. every 6 hours). When an Atlantic tropical cyclone isrelatively close to the U.S., these blackout times will be morefrequent, in order to accommodate the additional NHC Advisories and toperform other necessary duties. In addition, some or all of the parts ofthe website may close from time to time on an unscheduled basis, inorder to incorporate new information that is occasionally provided byNHC at times other than the usual Advisory schedules. These additionalblackout periods will also allow time for the new information to bedisseminated to interested parties. In one instance, the lengths ofwebsite blackout times will depend on the speed with which the NHCadvisories can be obtained, and their information transformed to updatedhurricane probabilities for the Hurricane Pools. Even when there are noAtlantic tropical cyclones in existence, financial investment unitprices will preferably be updated during the regular (e.g. 6-hourly)blackout periods, with decreasing of the early-entry discount by a smallamount.

When choosing to purchase financial investment units for a particularcounty or neighboring counties, it is preferable to encourage marketparticipants to thoroughly familiarize themselves with as much data onhurricanes as is publicly available. To assist in this effort, helpfulweather-related links may be provided on the Hurricane Pools website, aswell as other educational information that might be helpful. Forexample, a graphical history of storms which have occurred during thelast 113 years and their associated tracks may be provided to marketparticipants and other participants.

With reference to FIGS. 24-40, further details concerning the websiteare considered. In FIGS. 24-40 further examples of systems and methodsaccording to principles of the present invention is shown employing oneor more graphical user interfaces. As will be appreciated by thoseskilled in the art, the graphical user interfaces may be constructedusing known computer programs and programming languages. In general, thegraphical user interfaces are employed to facilitate investment relatedactions by a user, such as investment research, trial calculations andother activities and implementing financial investment decisions. Thegraphical user interfaces include, in addition to descriptive andinformation data, active graphical items, such as “buttons” having aconventional shape (e.g. round, oval) as well as special shapes (e.g. acounty map or a storm track).

In the examples given, the financial activity relates to naturallyoccurring events of the types mentioned herein, such as tropical stormactivity. As will be seen, it is generally preferred that the financialactivity be related to defined geographical areas (herein referred to as“regions”), represented in map or table form. For example, the regionsdefined in various embodiments of the present invention may correspondto well-known geographic or political boundaries, such as county,township or parish borders. However, it is sometimes expedient to definea region as an arbitrary geographical area, unrelated, or only looselyrelated to established boundaries. As will be appreciated, althoughregions having the approximate size of counties or parishes arepreferred in the illustrated examples, regions can be of virtually anysize desired, smaller than a county or alternatively, larger, such asgroupings of several counties. Further, although the illustratedembodiments are concerned with coastal regions of the United States,virtually any land are could be used as well. Also, although theillustrated embodiments pertain to first land strike of hurricane stormson the continental U.S., other natural peril events, such as theearliest official designation of a designated area being flooded, arealso contemplated.

Turning now to FIG. 24, a system and method for financial activityemploying a graphical user interface is shown illustrating anintroductory page generally indicated at 610. The graphical userinterface is the preferred way to practice the First Landfall Marketdescribed herein that includes a novel price-setting algorithm. Includedare active graphical selection item in the form of buttons 612 ofvarious text and graphic types. The buttons 612 provide links to afurther detailed entry page illustrated in FIG. 25. Also shown in FIG.24 are descriptive data components of the graphical user interfaceindicated at 614 and linking components indicated at 616 that invokeoutside resources that are external to the system described herein. Ifdesired, the linking components could be internalized to the systembeing described, if the system operator should choose to undertake thatadditional responsibility.

Referring now to FIG. 25, an entry page is shown. Included are activegraphical selection items or buttons 620, 622. Preferably button 620 isautomatically selected when the entry page is loaded, although button622 or neither button could be initially selected. In the preferredembodiment button 620 activates a data display showing probabilities forvarious land regions in real time. Included in the graphical userinterface, is a passive or static component in the form of a map 630 ofthe U.S., including eastern and southeastern coastal areas divided intoregions 632. These regions are overlaid by active graphical selectionitem or buttons 634 corresponding to the shape of the correspondingregion 632. Also included is a component of the graphical user interfacegenerally indicated at 638. This component is provided in the form of atable, with regions listed in one column, and their correspondingprobabilities listed in the adjacent column. The tabular component 638is preferably populated with active graphical selection item or buttonsin the form of text table entries, which serve as alternative activegraphical selection items to the graphical buttons 634.

Also included in the entry page of FIG. 25 are data components thatprovide information to the user. For example, data component 624provides a textual description of the real time map and relatedinformation displayed. Data component 626 provides textual descriptionof the current natural peril event status. At the time shown, a quietcondition is indicated with no ongoing storm having begun and no stormshaving occurred for the time period of the financial activity. Thepresent invention contemplates that the time period of the financialactivity may be defined in any manner desired. For example, the timeperiod could be the current calendar year, or the storm season for thecalendar year (e.g. June to November). Also contemplated are a number ofconsecutive time periods referred to herein as “events,” defined hereinas beginning at the year or season and ending, for example, with thefirst landfall of the year (defined according to preselected rules), andbeing immediately reset upon first landfall to start a second event forthe year, and so forth.

Referring now to FIG. 26, operation of button 634 is illustrated. Asmentioned, button 634 comprises one example of an active graphicalselection item. The button 634 may be activated as desired, by a keypress for example, or by merely moving the cursor over the button (anintuitive operation, owing to the graphical content of the button).Indicated in FIG. 26 is activation by moving the cursor over a button634 corresponding to a particular region defined by the financialactivity, corresponding to the geographical boundary of Charleston, S.C.The graphical user interface preferably responds in at least two ways.First, an information component in the form of a pop-up message 644appears next to the selected region.

The pop-up message 644 can contain virtually any desired messagecontent. Preferably, the message contains the probability of a firstland strike for the real time indicated in data component 624. Alsoincluded in pop-up message 644 is other data related to the financialactivity, such as the text name of the region selected and its inclusionin a regional grouping larger than county. As mentioned throughout thefigures and the examples of preferred embodiments, the probability offirst land strike is monitored by the facilitator of the financialactivity and is made available to the participants in a number ofdifferent ways. In place of the first land strike probability or inaddition thereto, the financial activity could monitor disasterestimates, in real time or on a historical basis, for example. Suchestimates could be calculated from meterological and other types ofdata, but could also be obtained from risk modeling companies and damageestimate services. For the selection illustrated in FIG. 26, the regionselected happens to correspond to a single county, and not part of alarger-sized geographical portion, and accordingly is designated as an“Individual County.” Also, characterization data related to theprobability is given in the pop-up message, indicating that the basisfor the probability as “historic,” although other bases could beemployed as well. The probability can be calculated in real time, inresponse to the user selection, or alternatively, can be extracted froma look-up table of previous calculations.

In the preferred embodiment, the graphical user interface responds tothe region selection by adjusting the tabular component 638 to displaythe text button corresponding to the chosen region in the middle of thetable, for ready contextual reference. In the embodiment illustrated,entries in the tabular component 638 are ranked according to decreasingprobability values, although virtually any ordering or arrangementdesired could be employed as well. Also, in the preferred embodiment,the same result could also be obtained by selecting the region's textualbutton 636, as opposed to the graphical button 634.

In the preferred embodiment, by selecting the desired region with amouse press, the graphical user interface responds by enlarging aportion of the display showing the selected region and its surroundingregions, in the manner shown in FIG. 27. In this figure, the selectedregion is Palm Beach, Fla., designated by the financial activity as an“Individual County,” indicating that the region corresponds to thegeographical and/or political boundary of that county of the State ofFlorida. The pop-up message 644 appears, as before, with information forthe region selected, and the textual button 636 is highlighted in thetabular component 638, and centered in that table area. If desired,additional buttons (not shown) can be provided to afford the user theopportunity to place an investment for the selected region, and/or toselect the neighboring regions, or to select additional individualregions.

Referring briefly to FIG. 25, button 622 is provided so that the usercan select financial activities related to storm tracks and individualstorms that have been recorded, with their related data analyzed to theextent necessary. As indicated herein, the U.S. government providesofficial determinations relating to storm events. For example, theNational Hurricane Center publishes ongoing reports of specific stormactivity, starting when a storm is a tropical depression at sea,continuing when a storm increases in intensity to become a tropicalstorm, and then when the storm further increases in intensity so as tobe designated a hurricane. Further, when a storm approaches land, e.g.is within 24 hours (estimated) of reaching landfall, official advisoriesare published at regular intervals, and at the appropriate time,official forecasts are also published. This data is, by its nature,historic, although data points along the course of a storm (the storm“track”) are given in the context of real time of near-real timeannouncements, advisories and predictions. Typically this data iscarefully analyzed by one or more responsible government agencies, andis recorded for future use. This data is preferably made available tothe financial activity user in a comprehensible form that allows a userto digest and correlate large amounts of data, quickly and easily, usinggraphical and graphics-related tools.

Referring now to FIG. 28, the illustrated display is preferably invokedby selecting button 622 in FIG. 25. Included in FIG. 28 is the map 630of the U.S. Any storm tracks for the current year would be shown, alongwith their corresponding table entries (to be explained herein). At thecurrent time, there are no storms to report. Note that data component624 is now blank, indicating that a user selection must be made toproceed further. Also included is an active graphical selection item inthe form of a pull-down table 650 that is initially selected to thecurrent year.

If desired, any preceding year or other desired time (such as a seasonalperiod, a date range, a recurring date or month) provided by thefinancial activity) can be selected. For example, referring to FIG. 29,the year 2005 is selected. In addition to the map 630 of the U.S., thetabular component 638 is populated with a listing of names of stormsthat are reported for the selected year 2005. Each entry in tabularcomponent 638 is an active graphical selection item button used toinvoke the menu sub-selection illustrated in FIG. 30. FIG. 29 also showsa number of active graphical selection item buttons 666 in the graphicalform of storm tracks, i.e. paths of storms reported in the selected timeperiod (herein the year 2005). The same result is achieved by eitherselecting the storm track button 666 or the textual storm name buttonsin the tabular component 638 of FIG. 29.

Referring now to FIG. 30, in the preferred embodiment, for each storm anoption is provided to either “Load track and forecast advisories”according to text button 660, or “Explore probabilities for this storm”according to text button 662. Note that data component 624 now indicatesthe current selection of the storm “Katrina” in year 2005.

If text button 660 in FIG. 30 is selected, then the track and forecastadvisories are loaded for the selected storm in the selected timeperiod, and the display of FIG. 31 is presented to the user. In additionto showing the tabular component 638, with its indication of theselected storm, the storm tracks of FIG. 30 are replaced with a singlestorm track 670 for the selected storm, shown in an expanded (enlargedwidth) size, and with a plurality of data points 672, for the point intime indicated in pull-down table. The data points may include, forexample, official advisories, such as NHC advisories, other warning orinformation services provided by the NHC or other third partyinformation source. In the illustrated embodiment, NHC advisories arethe preferred data points.

If text button 660 in FIG. 30 is selected, then the track and forecastadvisories are loaded for the selected storm in the selected timeperiod, and the display of FIG. 31 is presented to the user. In additionto showing the tabular component 638, with its indication of theselected storm, the storm tracks of FIG. 30 are replaced with a singlestorm track 670 for the selected storm, shown in an expanded (enlargedwidth) size, and with a plurality of data points 672 for the point intime indicated in pull-down table. The data points may include a varietyof different data types, such as forecast advisories or other forecastproducts by the NHC or other responsible government agency. In thepreferred embodiment, the active graphical selection item table 676contains a list of storm advisories published by the NHC for theselected storm. Each storm advisory has a textual button 678, preferablycorresponding to available storm track data sets that have been reportedfor each advisory published. Initially, the first table item isselected, and the corresponding data points appear as active graphicalselection item buttons 672. When buttons 672 are selected, correspondingdata (not shown) is displayed, including, for example, current (i.e.current as of the time of the advisory) storm location, direction, pathvelocity, internal wind velocities and other storm relatedmeteorological and climatological data.

Any of the textual advisory name buttons 678 of FIG. 31 can be selected,as desired. Referring to FIG. 32, the sixteenth advisory for theselected storm has been selected, #016, occurring at Sat, August 27th 9AM. Preferably, the overall track outline has been retained, but thedata points along the track have been updated. For example, data point682 indicates the “present” location of the storm, now designated ashurricane Katrina. Data points 684 indicate past advisory data points,whereas data points 686 indicate forecast advisories. FIG. 33illustrates the user selection of data for the data point 682 for the(historically) current position of the storm. Selection of the activegraphical selection item button 682 causes data component 690 to bedisplayed. A similar operation is illustrated in FIG. 34 where forecastadvisory button 686 is selected, with the graphical user interfaceresponding by displaying information component 692. FIG. 35 shows the27th advisory data set that occurred approximately at the time oflandfall. Data point 694 shows the historically current location ofhurricane Katrina. Note that the 36 hour forecast advisory data point696 lies outside the storm track 670, indicating that the hurricanelevel was downgraded approximately 30 hours beyond the selected point intime (i.e. the time of advisory 27 of the 2005 storm Katrina).

FIG. 36 illustrates the path predicted for the storm at Sat, August 27th3 PM. Note the predicted track 670 stopping at the southern tip ofIllinois, slightly after the passage of 96 hours beyond the historicallycurrent time. This figure illustrates how forecast conditions, stormevent probabilities, vary during the course of a storm. In FIG. 36,strike probabilities are listed in tabular component 638 for thisselected point in time. If desired, any of the advisory time pointslisted in table 676 could be chosen, and in response, the graphical userinterface will prepare probability data. If desired, this data can beprepared for all regions available to a user, or the program can waituntil the user selects a particular region. Either the regionprobabilities can be calculated in real time, or previously calculatedvalues can be extracted from available databases or other recordsavailable.

FIG. 37 illustrates the probability of first landstrike for a selectedgeographical location or region, for a particular day and time inhistory, within the context of a given storm occurring in the chosenyear. An information component in the form of a pop-up message 710appears next to the selected region. Adjacent table 712 displays aranking of probabilities (and optionally, other data, not shown) forregions in play at the selected date and time in history. The table 712preferably displays a list of textual buttons representing the names ofthe regions whose data is accessible to the user.

FIG. 38 illustrates the probabilities of first landstrike for allregions in play, as of a chosen date and time when forecast dataavailable through the textual button 718, applies. As indicated, acomplete set of probability data is made available to a user who isstudying the investment conditions that existed for an actual storm,located at the position indicated at 720. Included in the data madeavailable to the user, is the National Hurricane Center 12 hour forecastadvisory, as indicated above, with reference to textual button 718.

As mentioned above with respect to FIG. 26, the graphical user interfaceoperating in real time, or near-real time, responds to the regionselection by adjusting the tabular component 638 to display the textbutton corresponding to the chosen region in the middle of the table,for ready contextual reference. In the embodiment illustrated in FIG.26, entries in the tabular component 638 are ranked according todecreasing probability values, although virtually any ordering orarrangement desired could be employed as well. As indicated in thesingle column listing of probabilities for corresponding regions, only asingle probability is associated with each region is contemplated forthe current storm season. Referring now to FIG. 39, multipleprobabilities (i.e. three) are displayed for each region.

The arrangement of FIG. 39 reflects a financial activity in whichseveral events are conducted for a given storm season. A number ofpossibilities exist. Preferably, at the beginning of the year, somesmall number (e.g. three) events are opened and operated concurrently.In the event of a first hurricane landfall, the first of these isclosed, and the remaining two continue—they are preferably not initiallyopened at that time, as set out in the following example where, as soonas the first landfall occurs in a given storm season, the first eventsis closed, and a second events is declared open. Under this latter modeof operation, the number of events offered by a financial activity for agiven storm season is determined by natural peril events, with thefinancial activity responding to each land strike in the mannerindicated. The table component 724 indicates that three events haveoccurred at the moment in time indicated. As mentioned, it is generallypreferred that this moment in time displayed is as close to real time aspossible. In FIG. 39, the probabilities for the first three events aretabulated in columns 730, 732 and 734. Other financial datacorresponding to each region listed in the table is also displayed. Forexample, a theoretical return on investment (assuming an immediate firststrike in the region of interest) is given in column 738, a currenttotal investment is given in column 740 and a per-financial investmentunit purchase cost is indicated in column 742. Additional data items canbe included, either in the spreadsheet style illustrated in FIG. 39, orin a pop-up or pull-down style, for example, as may be desired.

Turning now to FIG. 40, activities pertaining to a purchase of afinancial investment are illustrated. An information component in theform of a pop-up message 748 appears next to the selected region.Message 748 may include probability data, as indicated and/or additionalinvestment data, such as meteorological data. Table component 752supplies additional data as may be necessary. For example, in additionto providing a textual confirmation of the region selected, the currenttime, hypothetical rate of return (assuming an immediate strike in theregion selected) and financial investment unit cost are displayed to theuser. If the user should decide to proceed, the number of financialinvestment units to be purchased are entered, and the total cost for thetransaction is displayed. Alternatively, the user may enter a financialamount, and the programs returns the number of financial investmentunits that can be purchased for the designated amount. As a final step,the user activates button 756 to finalize the offer to purchase. If theuser wishes to explore operation of the financial activity, but alsowishes to make certain that an offer to purchase is not tendered, theuser can activate button 762, requiring the user to clear the “trialrun” status previously chosen, before proceeding with an actualtransaction. This causes a display component 763 to appear with avariety of information, relevant to the user's current proposedposition.

FIG. 41 shows display component 763 in greater detail. Preferably,display component 763 is provided in the form of a table displaying avariety of data relevant to the user's proposed position, should thedesignated number of financial investment units be purchased. Includedis a Landfall Instantaneous Return Calculation, assuming that the stormimmediately makes landfall. Following, is a calculation of a LandfallConditional Floor, setting out the amount the user can expect to receiveif the storm should eventually hit the designated geographical area. Avariety of secondary market information is also reported to the user,including the current value of the proposed investment in the secondarymarket, based on market trading. The bid price, asking price and numberof available financial investment units are also given.

The display component 763 is preferably made available to users when auser logons onto the system at a later date. The display component isupdated, either automatically, or in response to a key press to keep theuser informed of his position in the market. FIG. 42 shows a summarydisplay of the user's market status, assuming the user has invested in anumber of different positions, schematically indicated by displaycomponents 763 in FIG. 42. A total display component 770 resembles theindividual display components 763, except for showing a total of theirdata components.

21.5 Disposition of Hurricane Pool Assets when No U.S. LandfallingHurricane Occurs

It is generally preferred that a “null” event be created as analternative to a selected geographical region. When employed, the nullevent is an investment option provided as an alternative to allgeographical areas, and indicates the participant's prediction that nohurricane landfall (or occurrence of another natural peril event) willoccur for a given financial activity time period.

As mentioned herein, it is possible that an entire year or other periodof financial activity can pass without a landfall occurrence. Whilevarious arrangements can be provided under exchange rules, it isgenerally preferred to account for this possibility as an option choice.It is preferred that the option choice be presented for trade as if itwere an eligible county or other geographical region. Financial activityrules are preferred that provide for payouts to holders of the nullcounty only at the conclusion of a financial activity period, such as ameteorological hurricane season,” generally considered to be December 1of the current year. If no hurricane makes landfall in the U.S., duringthe hurricane season, for example, the null county holders for allevents or other activities of the season will preferably receive a payout from a portion of the monies collected.

It can happen that a Hurricane Pool will be closed by the imminentapproach of a hurricane which subsequently fails to pass over any U.S.land area, according to the definition of a “hit” used by the HurricanePools (e.g. Hurricane Ophelia (2005) would have been one such case). TheHurricane Pools need to have a clear and automatic rule for thedisposition of the assets of such Hurricane Pools. Some possibilitiesare:

1. Preferably, a fund closed by an imminent approach of a storm thatdoes not make landfall is, to reopen that events when it is clear fromNHC or other official reports that the storm in question poses nofurther threat.

2. Transfer all assets and financial investment unit ownerships to thenext Hurricane Pool. For example, if Hurricane Ophelia had closedHurricane Pool #3 in 2005, all investments made in Hurricane Pool #3would be transferred to Hurricane Pool #4. Financial investment units inHurricane Pool #4 would then include those reflecting previousinvestments in Hurricane Pool #4, in addition to those purchased byinvestments in (the now closed) Hurricane Pool #3. Assets of thecombined Hurricane Pool #4 would then have been paid to financialinvestment units for counties qualifying according to the path ofHurricane Rita, regardless of whether they represented investments inHurricane Pool #3 or Hurricane Pool #4.

3. Transfer the assets, but not the financial investment unit ownership,to the next Hurricane Pool. Here, the money invested in Hurricane Pool#3 for the hurricane that eventually turned out to be Ophelia would havebeen transferred to Hurricane Pool #4 and paid according to the track ofhurricane Rita, but financial investment units purchased in the originalHurricane Pool #3 in this example, would not qualify for anydisbursements.

4. Another rule will be needed to govern assets of Hurricane Poolsremaining open at the end of the year. If desired, one of the aboveoptions, (although not necessarily the same as that governing mid-seasonHurricane Pool closures) or perhaps, some other option could be chosenfor pool disposition.

5. The pools can be returned to participants, less management costs, ifdesired.

6. Preferably, however, an additional outcome, called “null” event isdefined. For the Kth events, this corresponds, to the event whereinthere is no Kth U.S. hurricane landfall in the current year.Participants purchasing financial investment units in the null outcomesplit the pool of money at the end of hurricane season, if that eventsis still open as a consequence of a fund remaining open until that time.

21.6 Detailed Consideration of the “Hurricane Pools” Example

I. Introduction

The following is a detailed consideration of the Hurricane Pools examplegiven above. In one example, the number of Hurricane Pools in whichplayers may participate varies from year to year. Initially, K HurricanePools are opened, beginning on 1 January of the year before thehurricane season in question. Each of these relate to one of the first Ktropical cyclones passing over a portion of the U.S. with at leasthurricane-force winds, as determined by the National Hurricane Center orits successors, during the hurricane season (1 June through 30November). In relatively active years, additional Hurricane Pools may beopened as the hurricane season progresses. The Hurricane Pools enteredin a given natural peril event, less portions for state participationand Hurricane Pool management, are paid to entrants according to aformula that accounts for the chances of each hurricane striking thequalifying county(s), as assessed at the time a participant entry ismade.

21.7 Structure of the Hurricane Pools

A. One Example of Hurricane Pool Financial Investment Units

Entries (investments) in the Hurricane Pools may be made for individualcounties or other geographic regions (including the null event, whenpresent in the financial activity), in the form of “financial investmentunits.” The price of a financial investment unit varies in one instance(preferably at an initial stage), according to the probability of ahurricane strike on the county for which the entry is made, usinginformation available at the time the entry is made, and optionallymodified by a discount factor that encourages early entries andpenalizes later entries. Financial investment units may be pricedrelative to a benchmark, or “par” value, defined by an entry for themost vulnerable county historically, (Palm Beach, Fla.) made at thebeginning of the hurricane season (1 June). Choice of Palm Beach Countyas the par level is arbitrary because all of the financial investmentunit values are relative; but this choice may have market participantappeal, in that the prices of entries for all other counties will thenreflect an apparent discount.

The reference probability for a hurricane strike on Palm Beach county,as well as reference probabilities for the other n counties for whichentries are accepted (especially at an initial stage of marketactivity), is preferably derived from a climatological analysis of U.S.landfalling hurricanes that occurred from the late 19^(th) centurythrough the present. This type of analysis can specify the probabilitythat the center of a hurricane-force tropical cyclone will pass within75 nautical miles (86.25 statute miles) of the county center in a givenyear. These values can be adjusted for the size of a county byestimating the probability that a hurricane will track through thecounty in a given year, assuming that the county has a circular shape.Defining the probability of passage within 75 n. mi. of county i asQ_(i), the size-adjusted annual climatological hurricane strikeprobability is, according to the following Equation (1):

$\begin{matrix}{{Q_{i}^{*} = {{Q_{i}\frac{2\left( {A_{i}/\pi} \right)^{1/2}}{2 \cdot 87.25}} = {0.006541\mspace{14mu} A_{i}^{1/2}Q_{i}}}},} & (1)\end{matrix}$where A_(i) is the area of the county, in square statute miles. The87.25 mile counting radius is used to smooth the somewhat erratichistorical record of hurricane tracks.

Referring now to FIG. 19 a schematic diagram indicates a preferredtreatment of a geographical unit, herein, Palm Beach county, FL, with anarea of approximately A=2230 sq. mi., represented by a circle of thesame area (dashed). Hurricane centers passing within 86.25 miles of thecounty center (long arrows) have a probability of about 0.31 (ratio ofthe dashed circle diameter to 172.5 ml) of passing through the countyitself.

FIG. 19 illustrates the geometry behind Equation (1), for the case ofPalm Beach County. The area of this county is approximately A=2230 sq.mi., and the annual probability of a hurricane track within 86.25 mi. ofthe county center is Q=28.74%. Many storms tracking within this distanceof the county center will fail to pass through the county, but theproportion that will do so is given approximately by the ratio of thediameter of the circle approximating the county (=2[A/□]^(1/2)=53.3 mi.)to twice the search radius defining Q, or 172.5 mi. Therefore, for thiscounty, Q*=28.75% (53.3/172.5)=8.88%. The counties included in the FirstLandfall Market activity, their approximate areas, and their Q and Q*climatological values are calculated and made available for futurereference.

The reference probability for a hurricane strike on Palm Beach county inany single Hurricane Pool is smaller than Q*/100=0.0888, because thereare more than one U.S. hurricane landfalls in an average year. Thisaverage number of U.S. hurricane landfalls is □=1.7 hurricanes/year, sothe Palm Beach County reference probability is, according to thefollowing Equation (2):

$\begin{matrix}{p_{ref} = {\frac{Q^{*}}{100µ} = {\frac{8.88\%}{(100)(1.7)} = {0.0522.}}}} & (2)\end{matrix}$

In addition to depending on hurricane likelihoods for a county ofinterest in relation to the reference probability for Palm Beach countyin Equation (2), financial investment unit prices also increasegradually through the time period that entries are accepted, accordingto daily compounding of an annualized discount rate D that is multipliedin the financial investment unit pricing formula, according to thefollowing Equation (3):

$\begin{matrix}{{{Time}\text{-}{of}\text{-}{entry}\mspace{14mu}{adjustment}} = {\left( {1 + D} \right)^{\frac{{j\mspace{11mu}{date}} - 152}{365}}.}} & (3)\end{matrix}$Here jdate is the Julian date (consecutive numbering of the days of theyear), so that the exponent in Equation (3) is zero, and Equation (3)produces no change in the financial investment unit price, for 1 June(jdate=152). Julian days in the year prior to the hurricane season inquestion are negative. If the annual discount rate is 5%, then D=0.05.

Adjusting financial investment unit prices by multiplying by Equation(3) rewards early entries and penalizes late entries, in part as acompensation for opportunity costs. An appropriate value for thediscount rate D needs to be determined, and might be varied from year toyear to reflect values in then current financial markets. However, Dshould also include a very substantial premium over short- ormedium-term interest rates in order for this factor to have asignificant effect on financial investment unit prices, and so toencourage contributions to the Hurricane Pools well in advance of thebeginning of hurricane season. Referring now to FIG. 20, a graphicalplot shows financial investment unit price, relative to par on 1 June,for five values of the annual discount rate, D. FIG. 20 indicates valuesof Equation (3) as a function of date of entry into the Hurricane Pool,for a range of values of D. The relative financial investment unit priceis 1.0 for all discount rates at the par date of 1 June. For currentmarket rates on short-term money (D≈0.04, or 4%) the effect on financialinvestment unit price, shown by the dashed line, is negligible. Anannual discount rate of D=4.0 (i.e., 400%) is necessary to produce (forexample) a price differential of approximately 15% between 1 May and 1June.

The price per financial investment unit for a particular county, i, isdetermined by the probability of a hurricane strike on that county,p_(i), at the time the entry is made; and in relation to the par valuefor an entry on Palm Beach county (Equation 2) as of 1 June (Equation3). These factors are combined to determine the financial investmentunit price using the following Equation (4):

$\begin{matrix}{{{Price}\mspace{14mu}{per}\mspace{14mu}{share}} = {{F\left( {1 + D} \right)}^{\frac{{j\mspace{11mu}{date}} - 152}{365}}{\frac{\ln\left( {1 - p_{i}} \right)}{\ln\left( {1 - p_{ref}} \right)}.}}} & (4)\end{matrix}$This equation is the basic pricing tool for the Hurricane Pools. Here Fis an arbitrary pricing factor, that could be chosen according tomarketing considerations. It is the par price for one financialinvestment unit, for Palm Beach County on 1 June. For example, F=1corresponds to $1/financial investment unit. A higher pricing factor,such as F=100 ($100/financial investment unit) might have the effect ofsubtly encouraging some participants to enter more money in theHurricane Pool. The second factor in Equation (4) specifies that entriesmade before 1 June will be cheaper, and entries made after 1 June willbe more expensive, as indicated in Equation (3) and in FIG. 20. FIG. 21,table 1, shows an example of illustrative financial investment unitprices, in round numbers, for a range of strike probabilities p_(i),assuming purchase on. June, with F=$100/financial investment unit.

Finally, Equation (4) indicates that the financial investment unit pricefor county i depends on the probability p_(i) relative to the referenceclimatological probability p_(ref) for Palm Beach county, through thefunction −ln(1−p). This functional form has been chosen in order toobtain financial investment unit prices that are economically logical,particularly toward the extremes of the probability range. For p_(i)=0,Equation (4) produces a zero financial investment unit price: financialinvestment units in a county are free if there is absolutely no chancefor the Hurricane Pool to pay off for that county, and financialinvestment units are extremely cheap for counties where theprobabilities of being affected by hurricanes (e.g., in west Texas) arevanishingly small. At the other extreme, the financial investment unitprice approaches infinity as the probability that a hurricane willaffect the county approaches 1, so that the Hurricane Pools offer noreward for betting on a sure thing. FIG. 21, table 1 shows thedependence of financial investment unit prices (purchased on 1 June,with F=$100/financial investment unit) on the strike probability p_(i),for a few illustrative cases. According to equation (4), unless thereare Atlantic tropical cyclones currently in existence, counties forwhich the climatological probability Q_(i)*/(100μ)=p_(i)<p_(ref) (i.e.,all counties except Palm Beach), the price per financial investment unitwill be less than F, and accordingly most participants will receive adiscount in the purchase price.

21.8 Determination of Strike Probabilities for Equation (4)

The hurricane strike probabilities p_(i) in Equation (4) are based onthe best information available at any given time, that can reasonably beobtained in an automated way by the Hurricane Pool's website software.If no Atlantic tropical cyclones are in existence at the time of aHurricane Pool entry, that best information will be the unconditionalclimatological probability of a hurricane strike on the county inquestion for the k^(th) Hurricane Pool. These will be referred to in thefollowing as “Stage I” probabilities.

Sharper, that is, more detailed, or more accurate or better informedprobability information about hurricane strikes can be obtained when oneor more Atlantic tropical cyclones are in existence, but are too farfrom the U.S. for probability forecasts of hurricane-force winds forcounties of interest to be issued by the TPC. In such cases, theprobabilities p_(i) in Equation (4) are obtained from climatologicalvalues for each county, conditional on the existence of a tropicaldepression, or tropical cyclone such as a hurricane, in a given sectorof ocean. These conditional climatological probabilities will bereferred to as “Stage II” probabilities. In another example, pricing maybe determined solely, or partially by market activity, in combinationwith other factors described herein.

Finally, the TPC issues probability forecasts for hurricane-force windsduring the upcoming 5 days. These forecasts provide the “Stage III”probabilities, when they extend over land areas of interest, especially,when they predict landfall within the next 24 hours. To the extent thatthere may be more than one Atlantic tropical cyclone in existence at agiven time, Stage II and/or Stage III probabilities for each need to becombined in order to evaluate p_(i) in Equation (4). In one example,Stage I pricing is determined according to the probabilities and otherconsiderations provided herein, whereas Stage II and Stage III pricingis determined solely by market activity.

A. Stage I Probabilities

The Stage I probabilities are obtained from climatological values, in away that follows Equation (2) for the reference strike probability forPalm Beach county. In Equation (2), the size-corrected annual strikeprobability Q* is divided by the average number of strikes per year, μ,to reflect the fact that more than one hurricane affects the U.S. peryear, on average, and is converted from percentage to fractionalprobability. The Stage I probabilities are further corrected to reflectthe fact that, for the second and subsequent Hurricane Pools, it is lesslikely for there to be a corresponding U.S. hurricane landfall. That is,entering the first Hurricane Pool is less uncertain than is entering thesecond Hurricane Pool with respect to the Stage I probabilities, and sothe financial investment unit prices for the first Hurricane Poolsshould be higher. Similarly, the financial investment unit prices shouldbe higher for the second Hurricane Pool than for the third and anysubsequent Hurricane Pools. These adjustments are included in thecalculation of the Stage I probabilities using probabilities fordifferent numbers of landfalling hurricanes, as calculated using thePoisson distribution. This distribution is a conventional andwell-accepted probability model for allocating probability among thepossible numbers of hurricanes in a given year when the average numberper year is μ. Specifically, the Poisson probabilities for each possiblenumber, X, of U.S. landfalling hurricanes are, according to Equation(5):

$\begin{matrix}{{{\Pr\left\{ {X = x} \right\}} = \frac{\mu^{x}{\mathbb{e}}^{- \mu}}{x!}},\mspace{14mu}{x = 0},1,2,\ldots} & (5)\end{matrix}$Using these Poisson probabilities with μ=1.7 U.S. landfalling hurricanesper year, on average, the Stage I probabilities for the i^(th) county ink^(th) Hurricane Pool are, according to the following Equation (6):

$\begin{matrix}{{p_{i}^{(I)} = {\frac{Q_{i}^{*}}{100\;\mu}\frac{\Pr\left\{ {X \leq k} \right\}}{1 - {\Pr\left\{ {X = 0} \right\}}}}},\mspace{14mu}{i = 1},\ldots\mspace{14mu},{n.}} & (6)\end{matrix}$When a Stage I probability is the appropriate risk estimate for countyi, p_(i) ^((I)) is substituted for p_(i) in Equation (4) to determinethe financial investment unit price. For the k=1^(st) Hurricane Pool,the ratio of Poisson probabilities in Equation (6) is 1, so thatEquation (6) for County i is exactly analogous to Equation (2) for thereference county, Palm Beach. That is, p_(ref) in Equation (2) isnothing more than the Stage I probability for Palm Beach county in thefirst Hurricane Pool. For the second and subsequent Hurricane Pools,these Stage I probabilities are reduced to reflect the fact that thecorresponding hurricanes are less likely to occur. The purpose of thissecond factor in Equation (6) is to provide a further price advantage toearly entrants in the second and subsequent Hurricane Pools, which maynot pay off at all, relative to entrants who wait until after theformation of what may become the k^(th) landfalling hurricane beforeentering. FIG. 22, table 2 shows Poisson probabilities from Equation(5), calculated with μ=1.7 landfalls/year, the corresponding cumulativeprobabilities Pr{X≦x}, and the ratio of probabilities on the right-handside of Equation (6). FIG. 22, table 2 shows Poisson probabilities forμ=1.7 hurricane landfalls per year, with corresponding cumulativeprobabilities and ratios of probabilities used in Equation (6).B. Stage II Probabilities

When an Atlantic tropical cyclone is in existence, the Stage IIprobabilities p_(i) ^((II)) associated with county i being affected by ahurricane may increase from the respective Stage I value, depending onthe location and intensity of the storm. These Stage II probabilitiesare obtained by combining the Stage I probabilities, with conditionalclimatological relative frequencies of hurricane-force winds occurringwithin 120 n.mi. (138 statute miles) of each county center, given that atropical cyclone that is or will eventually become a named storm (i.e.,at least tropical storm strength) exists in one of 406 2.5 by 2.5 degreeregions of the Atlantic ocean, Caribbean Sea, or Gulf of Mexico. Theseconditional relative frequencies denote one of these ocean regions inwhich there is a tropical cyclone as j, and the conditional probabilitythat hurricane force winds due to this storm will eventually occurwithin 120 n. mi. of the center of county i as Q_(i,j). That is, foreach ocean region j, there is a data table similar to that for theunconditional climatological values Q_(i), although the conditionalQ_(i,j) climatological values are calculated with a larger smoothingradius (120 vs. 75 n. mi.) because there are fewer storms from which tocalculate the conditional relative frequencies. Accounting for thislarger smoothing radius, the size-adjusted conditional relativefrequencies Q*_(i,j) are calculated, analogously to Equation (1), usingthe following Equation (7):

$\begin{matrix}{{{Q_{i,j}^{*} = {{Q_{i,j}\frac{2\left( {A_{i}/\pi} \right)^{1/2}}{2 \cdot 138}} = {0.004088\mspace{14mu} A_{i}^{1/2}Q_{i,j}}}};}{{i = 1},\ldots\mspace{14mu},{n;\mspace{14mu}{j = 1}},\ldots\mspace{14mu},406.}} & (7)\end{matrix}$Stage II probabilities are computed by combining these area-adjustedconditional relative frequencies with the corresponding Stage Iprobabilities, according to the following Equation (8):p _(i) ^((II)) =p _(i) ^((I)) +q _(i,j) −p _(i) ^((I)) q _(i,j), i=1, .. . , n; j=1, . . . , 406;  (8)where, according to the following Equation (9):

$\begin{matrix}{q_{i,j} = \left\{ \begin{matrix}{\frac{{.837}Q_{i,j}^{*}}{100},} & {\begin{matrix}{{if}\mspace{14mu}{the}\mspace{14mu}{storm}\mspace{14mu}{in}\mspace{14mu}{ocean}\mspace{14mu}{sector}\mspace{14mu} j} \\{{is}\mspace{14mu} a\mspace{14mu}{unnamed}\mspace{14mu}{depression}}\end{matrix}} \\{\frac{Q_{i,j}^{*}}{100},} & {\begin{matrix}{{if}\mspace{14mu}{the}\mspace{14mu}{storm}\mspace{14mu}{in}\mspace{14mu}{ocean}\mspace{14mu}{sector}\mspace{14mu} j} \\{{is}\mspace{14mu} a\mspace{14mu}{tropical}\mspace{14mu}{storm}}\end{matrix}} \\{\frac{1.72Q_{i,j}^{*}}{100},} & {\begin{matrix}{{if}\mspace{14mu}{the}\mspace{14mu}{storm}\mspace{14mu}{in}\mspace{14mu}{ocean}\mspace{14mu}{sector}\mspace{14mu} j} \\{{is}\mspace{14mu} a\mspace{14mu}{{hurricane}.}}\end{matrix}}\end{matrix} \right.} & \begin{matrix}\left( {9a} \right) \\\; \\\left( {9b} \right) \\\; \\\left( {9c} \right) \\\;\end{matrix}\end{matrix}$Here, 0.837 is the proportion of tropical depressions that have gone onto at least tropical weather strength (1991-2004, reflecting currentoperational practice at NHC), and 1.72 is the ratio (1886-1998) of thenumbers of tropical storms to hurricanes in the Atlantic basin. Thepurpose of Equation (9) is to reflect the fact that the existence of ahurricane is more threatening, on average, than the presence of atropical storm, which is in turn more threatening than the presence of atropical depression. The probability from Equation (8) is substitutedfor p_(i) in Equation (4) when the Stage II risk assessment isappropriate for county i.

Equation (8) reflects the increase in risk, over and above the baselinerisk to county expressed by p_(i) ^((I)), attributable to the presenceof a tropical cyclone in ocean sector j. The conditional probabilitiesq_(i,j) are combined with (rather than replace) the Stage Iprobabilities in Equation (8), because county i continues to be at(climatological) risk for being struck by a hurricane, even if the stormin ocean sector j fails to make landfall as a hurricane. If theconditional probability q_(i,j) is substantial, p_(i) ^((II)) will beappreciably larger than p_(i) ^((I)). If the tropical cyclone in oceansector j has negligible probability of affecting county i as ahurricane, Equation (8) implies p_(i) ^((II))≈p_(i) ^((I)).

C. Stage III Probabilities

Stage III probabilities will be based on the NHC hurricane windforecasts provided as part of the official advisory for each tropicalcyclone. The system that is expected to be in place for these forecastsfor the 2006, 2007, and later hurricane seasons (currently described byTPC as “experimental”) will produce probability forecasts for windspeeds of at least hurricane force within the upcoming 120 hours (aftereach advisory), when these probabilities are at least 2.5%. Examples areshown at www.nhc.noaa.gov/feedback-pws-graphics2.shtml?.

In the current TPC forecast, hurricane-force winds in county i aredenoted as f_(i). Analogously to Equation (8), the Stage IIIprobabilities are computed by combining these forecasts with the Stage Iprobabilities from Equation (6), again because failure of the currenttropical cyclone to affect the U.S. as a hurricane does not preclude thek^(th) Hurricane Pool from paying out for some subsequent storm.Specifically, the Stage III probabilities are computed using thefollowing Equation (10):p _(i) ^((III)) =p _(i) ^((I)) +f _(i) −p _(i) ^((I)) f _(i),f_(i)>q_(i,j),  (10a)orp _(i) ^((III)) =p _(i) ^((II)), f_(i)<q_(i,j).  (10b)Equation (10b) includes the possibility that the storm in question mayaffect a county for which f_(i)=0, because these TPC forecasts are setto zero if the probability is smaller than 2.5%. Again the Stage IIIprobabilities from Equation (10) are substituted for p_(i) in Equation(4) when explicit TPC forecasts for hurricane-force winds are currentfor some portion of the U.S.D. Combining Stage II and Stage III Probabilities

It can happen that two or more Atlantic tropical cyclones are inexistence at the same time. In such cases, their strike probabilitiesfor each county i need to be combined in some way, to yield the largerprobability that either one or the other might affect the county inquestion. Let p_(i)(1) be the probability of the first of these stormsfor county i, calculated using either Equation (8) or Equation (10), asappropriate. Similarly, let p_(i)(2) be the corresponding value for thesecond of these storms. If there are only two such cyclones present, thecombined probability p_(i) to be used in the pricing Equation (4) isobtained using the following Equation (11)p _(i) =p _(i)(1)+p _(i)(2)−p _(i)(1)p _(i)(2).  (11)This probability would be applied equally to the next two HurricanePools (assuming that there are two or more) that are still active andaccepting entries.

If there is a third such tropical cyclone, denote its probability forcounty i, calculated from the Equation appropriate to its Stage, asp_(i)(3). The combined probability for county in Equation (4) would thenbe, according to Equation (12)

$\begin{matrix}{p_{i} = {{p_{i}(1)} + {p_{i}(2)} + {p_{i}(3)} - {{p_{i}(1)}{p_{i}(2)}} - {{p_{i}(1)}{p_{i}(3)}} - {{p_{i}(2)}{p_{i}(3)}} + {{p_{i}(1)}{p_{i}(2)}{{p_{i}(3)}.}}}} & (12)\end{matrix}$This probability would be applied to the next (up to) three HurricanePools still accepting entries.21.9 Closing Hurricane Pools

Hurricane Pools cease to be available for further entries when thecorresponding hurricane is sufficiently close to a U.S. land area.“Sufficiently close” could mean that either a hurricane watch orhurricane warning has been issued for a U.S. coastal county. BecausePuerto Rico and the U.S. Virgin Islands are relatively far from the U.S.mainland, a Hurricane Pool can be closed for these two territorieswithout it necessarily being closed for the rest of the U.S. Because ofthe prevailing westward tracks of tropical cyclones at low latitudes, aHurricane Pool that is closed because of storm proximity to the U.S.mainland is also closed for Puerto Rico and the U.S. Virgin Islands.

21.10 Payout Algorithm

Financial investment units purchased in the k^(th) Hurricane Pool forcounties traversed by the k^(th) U.S. landfalling hurricane are“qualifying financial investment units.” In one instance, thesecountries are defined by the NHC operational adversaries. In anotherinstance, these counties are defined as those containing a “best track”hurricane position, or a portion of a line connecting two “best track”hurricane positions, as portrayed in the “best track” Table of theofficial TPC Tropical Cyclone Report for that storm. Payouts arepreferably determined by dividing the available Hurricane Pools (e.g.participant entries less state and management percentages) by the numberof qualifying financial investment units, and paying that amount foreach qualifying financial investment unit.

In cases where there may be multiple tropical cyclones in existence atthe same time, priority is determined according to time of first U.S.landfall. For example, if the hypothetical hurricane “Alice” makeslandfall after hurricane “Bob,” “Bob” would be assigned to Pool 1 and“Alice” would be assigned to Pool 2.

It is anticipated that payouts to Qualifying Financial investment unitswill be made within two weeks of the final NHC Advisory for the storm inquestion. In unusual cases, such as for storms that may have thepotential to reintensify and affect the U.S. again, Pool payouts may bedelayed beyond two weeks at the sole discretion of the PoolsAdministrators. In all cases, Pools disbursements will be made on thebasis of the best and most recent information available from theNational Hurricane Center at that time about the storm in question, andwill not be subject to revision in the event of subsequent updates tothat information. In the event, however, that there is no landfallofficially reported for the season or other time period, a “null” eventcan be provided as an investment option, as described herein.

21.11 Caveats Regarding Pool Payouts

The above rules for determining Qualifying Financial investment unitshave been somewhat idealized, relative to real-world hurricane behavior,in the interest of having a promptly available, clear, explicit andautomatic way of disbursing Pool assets to Qualifying Financialinvestment units. In particular:

-   -   There will often be counties experiencing hurricane-force winds        and/or other hurricane impacts that nevertheless do not qualify        as having been “hit” according to the definition used by the        Hurricane Pools. This will be the case especially for the larger        and more powerful storms. Pool market participants whose        intention is to, in effect, supplement their insurance coverage        will therefore be encouraged to invest in surrounding counties        also. To encourage market participants to protect themselves,        the Pools site will automatically flash several counties which        border the county initially selected and urge market        participants to spread their investment to include surrounding        counties. In this manner, which we call a “collar” the market        participant will have greater opportunity to collect pools if        damage occurs but the eye of the hurricane does not enter their        county.    -   Because qualifying counties are determined on the basis of storm        positions only at particular, and possibly irregular times,        small discrepancies between the calculated track (used to        determine Qualifying Financial investment units) and the actual        track (as determined some months later in the official NHC        Tropical Cyclone Report for that storm, or that might be evident        at the time of the storm from a events of weather-radar images,        for example) can and will occur. Again, it may be advisable for        some individuals to invest in nearby counties, in addition to        the county(s) in which they have the most interest.    -   The U.S. Census Bureau data files are only approximations to the        true geographical outlines of many counties. They consist of a        collection of line segments, and so will not accurately follow        curving county boundaries. In addition, portions of some        counties (particularly relatively small islands) are not        included in the Census Bureau's Cartographic Boundary Files. For        example, the Dry Tortugas are not included in the Cartographic        Boundary File for Florida, so that a hurricane passing over this        portion of Monroe County, Florida, would not by itself        constitute a “hit” on Monroe county for the purpose of        determining Qualifying Financial investment units.        21.12 Website Features

The Hurricane Pool will preferably be run through a website thatcalculates financial investment unit prices automatically, according toinformation from TPC advisories that are updated four times daily.Accordingly, it may be necessary for the site to be unavailable foraccepting entries for short, pre-scheduled, periods of time every 6hours for example. In addition, the website may need to be able to closefrom time to time on an unscheduled basis, in order to incorporate newinformation that is occasionally provided by the TPC at other than thescheduled 6-hourly update times. The lengths of these website blackouttimes, if any, will depend on the speed with which the NHC advisoriescan be obtained, and their information transformed to updated hurricaneprobabilities for the Hurricane Pool. Even when there are no Atlantictropical cyclones in existence, financial investment unit prices will beupdated during the regular blackout periods, by incrementing the date,jdate in Equation (4), by 0.25, four times daily.

Preferably, first-time entrants will need to register. Passwordprotection may be preferred if a single credit-card account is to beassociated with each registration, in order for any eventual payouts tobe made to that account. SSN information is preferably made part of theregistration in order for the IRS (and possibly also some states) totrack tax liability on any payout. As an alternative, an outside servicecan provide credit or other financial services.

Current financial investment unit prices (or other financial investmentunit) for all available counties are preferably displayed bothgraphically and in tabular form. Clickable maps (whole-coast, andindividual state) are also preferably made available, with financialinvestment unit prices indicated approximately with a color code.State-by-state pull down menus could also be provided if desired.Participants are preferably given the option of specifying their entrieseither in terms of financial investment units bought, or dollars to beentered, for each county selected.

Preferably, a whole-state entry can be defined by automatically issuingan equal number of financial investment units for each county in play,within the state in question. This approach would place more money oncounties more likely to be affected, and so would severely down-weightessentially zero-probability counties, such as those in west Texas. Herethe number of financial investment units bought for each county issimply the dollar amount to be entered, divided by the sum of financialinvestment unit prices according to Equation (4), over all counties inthat state.

The sums entered to date in each Hurricane Pool (and available forsubsequent payout) are preferably posted and continuously updated.However, it may be difficult to calculate for potential entrants thepossible payoffs for particular entries that they are contemplating,because those payoffs will depend on the track of the eventual storm inquestion. However, it is also possible to show a minimum payout or“floor” when such feature is desired.

As mentioned above, it is sometimes preferred to provide an expectationof a minimum payout or “floor” for participants that suffer damage froma natural peril event. The “floor” is a minimum payout, conditional onthe county or other geographical area in which an market participantholds an investment unit being “hit.”

21.13 Algorithm Parameters

As noted above, several parameters in the financial investment unitprice algorithm are adjustable. In one example these parameters could bedefined before the beginning of a given year's Hurricane Pools asfollows. These exemplary parameters are:

-   -   K=# of Hurricane Pools that will be opened initially.    -   n=# of counties in the game    -   D=discount rate (as discussed above)    -   F=pricing factor (as discussed above)

The choice for the number, K, of initial Hurricane Pools to be runinvolves a tradeoff between numbers of years when one or more HurricanePools do not pay off, versus numbers of years when there are more U.S.landfalling hurricanes than initial Hurricane Pools. Using the Poissonprobabilities from IG. 21 table 1, these tradeoffs are approximately asindicated in FIG. 23, table 3.

If all counties in an included state will be in play, it is necessaryonly to specify the states to be considered in order to determine n. Forexample Oklahoma has a single county with Q≠0, and Kentucky has seven.All eight of these have Q=0.01. Accordingly, it is preferred that thesestates not be included in the financial activity. A large number ofeffectively irrelevant counties may also be excluded under this plan,especially in Texas, but also in Arkansas and Tennessee.

XXII. First Landfall Market With Price-Setting Adaptive ControlAlgorithm

22.11. Introduction

As mentioned herein, a concept of a First Landfall Market has beenintroduced in which options or other investment units are made availablefor purchase at prices that change on an ongoing basis, once marketactivity commences. For example, before the first market activity,prices are preferably set using historical climatological probabilities.Thereafter, as purchases are made, prices are changed on an ongoingbasis. Financial activity is preferably carried out using one or morecomputer systems and a graphical user interface, preferably, the oneshown in FIGS. 24-49.

The present invention provides the ability to conduct a one-sided(buyers only), parimutuel market, over an indefinitely long period oftime. This ability is achieved through use of an adaptive controlalgorithm (described herein) which, through a self-updating process,“learns” the aggregate probabilities believed by the marketparticipants, as revealed through their buying decisions. Because pricesfor the financial instruments in this market are set in proportion tothese probabilities, this algorithm provides a rational and fairmechanism for price discovery in a one-sided market, in whichinformation available to the market participants may change over time.

In general, a parimutuel market is preferred for a number of reasons.Ordinary bilateral markets work well if there is a good balance ofbuyers and sellers, whose interaction provides a robust mechanism forprice discovery. However, in bilateral markets where there is asubstantial imbalance between buyers and sellers, market activity isstrongly constrained. Markets for catastrophic natural events, such asbut not limited to hurricanes, are prime examples of this difficulty,because most people will want to be buyers (who are “long” thecatastrophic event), in order that they can be compensated financiallyif the event occurs. There are few if any people to take the other sidesof those contracts (i.e., potential sellers), because there are few ifany potential market participants who will be hurt if the catastrophicnatural event does not occur. The result of an impasse such as this willbe a severe constraint on the liquidity of the market, so that itfunctions poorly or possibly not at all.

The problem of limited liquidity for markets like this can be approachedby creating a parimutuel market, in which all participants are buyers:it is “one-sided”. Because an overall parimutuel pool is split amonginvestors for the single specific outcome that eventually occurs,liquidity in this market is provided by investors in the outcomes thatdid not occur, even though all participants were “buyers.” In thecontext of the preferred embodiment described herein based on coastalregions, where the outcomes pertain to a large number of geographicregions where a hurricane might make landfall, the effect is thatliquidity is pooled over space in a single market, rather than therebeing many small markets in individual outcomes, each of which isstarved for liquidity.

One problem to be overcome in the structuring of a parimutuel market isthat of price discovery. Because these markets are one-sided (there areonly buyers), price discovery cannot be accomplished through theinteraction and negotiation between buyers and sellers. There is morethan one solution to the problem of price discovery in parimutuelmarkets. One approach is to hold an auction in which participants bid“limit prices” for a fixed payout that is contingent on the occurrenceof the event corresponding to the contract they are bidding on. The bidsare received and analyzed and as many orders as possible are filled,given the constraint that, whatever happens, there must be enough moneyin the parimutuel pool (composed of premia paid from the successfulbids) to satisfy any of the contingent claims. Price discovery isachieved by finding the bid level for each outcome that is sufficient tomeet this condition—bids that are too low are not filled. One limitationof this approach is that it will not function fairly if someparticipants have access to systematically better information thanothers. As a practical matter, this means that the auctions can remainopen for only very short periods of time, so that earlier bidders arenot disadvantaged by better information that may be available to laterbidders. Therefore, although liquidity can be pooled across the outcomes(e.g., pooled geographically for a hurricane market, if it were run asan auction), it is not pooled through time. Such a market can thereforehave liquidity problems that limit its successful functioning.

The present invention offers an advantage over this approach to pricediscovery in that it circumvents this problem, by dynamically updatingthe risks associated with each of the potential outcomes (e.g., theprobabilities that each coastal segment might be hit by the nexthurricane), as encoded (and empirically observed) by the marketparticipants' aggregate probabilities. Because prices are determined inproportion to these probabilities, they reflect the risks as perceivedin aggregate by the market participants at the time of purchase. Theresult is that the market can be run continuously, and for anarbitrarily long period, so that liquidity can be pooled through time aswell as over (geographical) space, thus improving the functioning of themarket relative to the limited-time mechanism of an auction activity,for example.

To summarize the market effects, when a contract for one of the outcomesis accepted (bought), the adaptive control algorithm described hereinautomatically increases its estimate of the market probability for theoutcome purchased (equation 85), and reduces the market probabilitiesfor all other outcomes (equation 86). Consider an “early” buyer for oneof the outcomes, when the available information is relativelyuninformative. Market probabilities for the various outcomes will berelatively small, because it is quite uncertain which might eventuallyhappen and market probabilities are spread quite broadly across allpossible outcomes. A buyer at this point will pay a relatively smallprice, because the market probability is low, and price is proportionalto the market probability. Now, as time goes on it may become more clearwhich outcomes are more favored to occur. More buying will occur forthese more favored outcomes, which will be reflected in progressivelyhigher market probabilities for them, and therefore higher prices forthem as time goes on. Later buyers of contracts for these outcomes willpay more than the earlier buyers, so that the earlier buyers are notdisadvantaged even though their participation was based on less preciseinformation.

22.2. Procedure for Constructing a Continuous One-Sided ParimutuelMarket

The following describes a preferred method for constructing a parimutuelFirst Hurricane Landfall Market in which there is one seller and manybuyers.

a. Define a set of mutually exclusive and collectively exhaustiveunknown future events, having significant economic consequences forpotential market participants. “Mutually exclusive” means that at mostone of these events can occur, and “collectively exhaustive” means thatat least one of these events must occur. For the preferred embodimentdescribed herein, directed to markets for hurricane landfalls, theseevents are coastal segments including location of the next U.S.hurricane landfall, or the event that there will be no further U.S.hurricane landfalls in the current market period (the “No Landfalls”)event.

Alternatively, in a market for earthquake occurrences, the outcomesmight be regions in which the epicenter of the next earthquake of atleast magnitude 6 will occur. In a market for tsunami occurrences theoutcomes could relate to coastal locations experiencing maximum tsunamiheights, given that it is higher than 2 m.

That the defined events have significant economic consequences is apreferred condition so that the markets described here are more readilyrecognized as legitimate financial markets, rather than gaming orgambling.

b. Define a timeframe over which the market will operate. For thepreferred embodiments directed to markets for hurricane landfalls, thisperiod is the calendar year, because there is a natural annual cycle inthe climatological occurrence of hurricanes, leading to the concept ofthe “hurricane season” extending from June through November for thenorth Atlantic basin. If no U.S. hurricane landfall occurs by the end ofa given calendar year, the outcome “No Landfalls” is deemed to haveoccurred.

Alternatively, in markets for such events as earthquakes or tsunamis,there is no such natural e.g. annual cycle on the basis of which tochoose the timeframe of the markets. In these cases, the timeframe forthe markets might be chosen as one decade. For example if no earthquakeof at least magnitude 6 occurs in any of the regions chosen for anearthquake market in Step 22.2a over the course of a decade, then the“No Earthquakes” outcome is deemed to have occurred.

c. Make a preliminary risk assessment for the outcomes in Step 22.2a,expressed in terms of an initial probability for each outcome. Thenature of this risk assessment will depend on the degree of scientificunderstanding of the (physical, economic, or other) phenomenonunderlying the market, and the length and quality of prior data relatingto this phenomenon. For example, for north Atlantic hurricanes,reasonably good historical data on location and intensity of U.S.landfalls are available from the mid- to late 19th century.

For other markets the available information on which to base an initialprobability assessment may be of lesser quality, quantity, or both. Inextreme situations no such information may be available, in which caseit may be desirable to initially spread probability equally among thepossible outcomes, an expedient that is not likely to substantiallyimpair the subsequent functioning of the market.

Denote these initial probabilities for the possible outcomes as π₀ ^(i).Here the superscript i indicates one of the events defined in Step22.2a, and the subscript 0 denotes that these probabilities pertain tothe initialization time for the market.

d. Define an overall magnitude for the pricing and expected payouts forthe outcomes. This magnitude will depend on the general level of thefinancial risks to be mutualized in the market, and is embodied in the“par” value of one contract in the market, denoted c. For example, inthe preferred embodiments of markets for hurricane landfalls, anappropriate choice for this parameter could be c=$1000. For markets inwhich the financial “stakes” are substantially lower than the magnitudeof economic damage inflicted by hurricanes, a smaller value for c wouldbe appropriate.

Having chosen an appropriate value of c for a particular market, pricesfor contracts in that market are set in proportion to the currentprobability for the outcome in question:P _(t) ^(i)=π_(t−1) ^(i) cexp[rj/365].  (83)

Where r is an annualized interest rate (or a constant proportionalthereto) and j is the relative julian date since the current marketopened. Accordingly, in the following, expressions that are given formarket or pricing probabilities can be directly converted intoexpressions of market prices and vice versa, using equation 83.

Here again the superscripts i indicate which of the outcomes from Step22.2a is being considered, and the subscript t indicates a time index(i.e. relative to its position in the sequence of purchases) at whichthe price is being calculated, based on the pricing probability π^(i)_(t−1) that was updated at the previous step. At the time of marketinitialization, P^(i) ₀ is the initial price for outcome i, and is basedon the initial market probability value π^(j) ₀. The factor exp [rj/365]in Equation 83 equalizes the value of investments made at differenttimes, in which r indicates an annualized interest rate (e.g., r=0.05for an annualized rate of 5%), and j indicates the number of days sincethe opening of the market.

e. Provide-initial liquidity to seed the market. The market begins withinitial liquidity provided to the Mutualized Risk Pool (MRP). It ispreferred that this amount is on the order of 100 to 1000 times largerthan the “par” value, c. In return for this initial investment, theseeding institution receives contracts for each of the outcomes definedin Step 22.2a. Because there is not yet a market consensus for theprobabilities of these outcomes, this initial stake is allocatedaccording to the preliminary probabilities, π^(i) ₀, reflectinghistorical risks or other relevant prior information. The result is thatthe initial number of contracts, n, in each outcome is equal, and givenby

$\begin{matrix}{n_{0}^{i} = {\frac{{MRP}_{0}}{c}.}} & (84)\end{matrix}$

For example, if the MRP is seeded initially with MRP₀=$1,000,000, andc=$1000, then the market begins with n^(i) ₀=1000 contracts in eachoutcome i. This allocation follows from Equation 83, with j=0.

f Open the market to investment, continually updating the outcomeprobabilities consistent with buying interest. The key element of pricedetermination in Equation 83 is the market (i.e., pricing) probabilityπ_(t) ^(i). At any given time t these reflect the market consensus forthe probabilities of each of the defined outcomes. Preferably, followingthe purchase of each additional new contract, the price for thatcontract is increased according to equation 85. The preferred increase,P^(i) _(t) is added to the MRP, and new prices for contracts for allother outcomes are recalculated, preferably lowered according toequation 86, to reflect a larger market consensus probability for theoutcome, i, that has just been purchased:π_(t) ^(i)=π_(t−1) ^(i)+α_(t)π_(t−1) ^(i)(1−π_(t−1) ^(i))  (85)Here α_(t) is the price adjustment rate parameter, 0<α_(t)<<1, to beexplained in more detail below, controlling the rate at which pricesadjust to buying activity. Because the probabilities for all outcomesmust sum to 1, market probabilities for the outcomes that were notpurchased in the most recent transaction are decreased proportionally:π_(t) ^(k)=π_(t−1) ^(k)(1−α_(t)π_(t−1) ^(k)), k≠i,  (86)

In the above equations, the “time” index t refers not to chronologicaltime, but rather to a counter tracking the sequence of purchases ofindividual contracts. So, for example, if a block of 100 contracts werebeing purchased for outcome i, Equations 85 and 86 would be iterated 100times during this process, each with a different time index value, t.The result would be that each of the 100 contracts in this block wouldcost slightly more than the previous contract in the block (throughoperation of. Equation 85)—the demand for contracts for outcome i wouldprogressively increase the price of subsequent contracts for thisoutcome (through the increase in probability defined by Equation 85),while depressing prices for the contracts for the remaining outcomes(through operation of Equation 86).

The probability updating procedures in Equations 85 and 86 are generallyrelated to a class of stochastic approximation algorithms (H. J. Kushnerand G. G. Yin. Stochastic Approximation and Recursive Algorithms andApplications. Springer-Verlag, 2003) although these equations representnew members of this class of algorithms. Proofs are given herein forconvergence of Equations 85 and 86 to the consensus of marketparticipants' probabilities for the outcomes, thus demonstrating thesuitability of these algorithms for pricing contracts in one-sidedmarkets in a way that is consistent with the risks as perceived at anygiven time by the market participants. That is, Equations 85 and 86automatically learn investors' probabilities for the outcomes inresponse to their collective actions in the market.

In the above, Equations 85, 86 reflect the pricing probability, π. Theseequations could be readily adapted to reflect Price, P, using Equation83.

g. Minimum pricing probability. As an optional step, to ensure numericalstability, preferred pricing probabilities are not allowed to fall below0.0001. Accordingly, the preferred minimum contract price is set at avalue such as 0.0001c, or $0.10 for c=$1000.h. Maintaining a guaranteed minimum payout. This optional step considersa situation where buying becomes very concentrated in one or a fewoutcomes before the MRP has accumulated sufficient liquidity, it ispossible for the payouts for contracts in that outcome to be dilutedsubstantially. Preferably, this dilution is not allowed to proceed belowa fixed, predetermined fraction of par, for example 50%, and if anadditional purchase would drive the potential payout below this levelthe price for that contract is increased sufficiently to cover theminimum payout. In one example, this price increase is set to 50% of parin this example, or $500 per contract if c=$1000.i. Determine which event has occurred, and disburse the MRP. At orbefore the end of the timeframe defined in section 22.2b, one and onlyone of the mutually exclusive and collectively exhaustive outcomes willhave occurred (since a null outcome is included, all possibilities arecovered). At that time, all money in the MRP is disbursed, in proportionto the number of contracts held for the outcome that did occur.Specifically, if outcome i occurs, each contract previously purchasedfor that outcome entitles the owner to

$\begin{matrix}{{{{Disbursment}\mspace{14mu}{per}\mspace{14mu}{contract}\mspace{14mu}{for}\mspace{14mu}{outcome}\mspace{14mu} i} = \frac{{MRP}_{\tau}}{n_{\tau}^{i}}},} & (87)\end{matrix}$where τ indicates the final time index t at which the event occurs.22.3. Considerations Regarding the Price Adjustment Parameter, α

The mathematical development of Equations 85 and 86, and the convergenceproofs presented herein, do not specify the magnitude of α_(t) otherthan it should be small (0<α_(t)<<1), and not decay too rapidly. Thusthe probability-updating algorithms in Equations 85 and 86 will convergeto market participants' aggregate probability distribution for theoutcomes, for any of a rather broad range of methods that are used tochoose this parameter. One simple choice is to assign a constant value,α_(t)=α=0.001, for example.

In accordance with one aspect of the present invention, at is allowed tovary with market conditions in order to improve the convergenceproperties of the algorithms in Equations 85 and 86. It is shown in thefollowing appendix that if this market structure is in balance, in thesense that the ratios of prices to indicative payouts for each outcomei, v^(i) _(t)=P^(i) _(t)/[MRP_(t)/n^(i) _(t)], are well reflected by thecorresponding pricing probabilities π^(i) _(t−1) used to compute P^(i)_(t), then the price adjustment parameter α_(t) in Equations 85 and 86that maintains this balance for the outcome i being purchased isα_(t)=1/ n _(t), where the average number of contracts per outcome issimply

$\begin{matrix}{{\overset{\_}{n}}_{t} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}n_{t}^{m}}}} & (88)\end{matrix}$Here M is the total number of outcomes defined in Step 22.2a. The resultis that prices will move more quickly for a given level of buyingactivity when the market is relatively thin ( n _(t) relatively small),and will move more slowly when the market is relatively well developed (n _(t) relatively large). However, in order to allow the market to reactquickly to imbalances, it has been found that α_(t) should in general belarger than 1/ n _(t), but should still decrease in proportion to n _(t)when n _(t) is sufficiently large:α_(t)=min [α_(max),H/ n _(t−1)]  (89)Here H=50 is an empirically determined parameter, and α_(max) is theinitial value for α_(t). Because the initial seeding has been allocatedconsistent with the initial probability estimates π^(j) ₀, the market isinitially in balance with the risks as expressed by these probabilities.Consequently,

$\begin{matrix}{\alpha_{\max} = {\frac{1}{{\overset{\_}{n}}_{0}} = {\frac{c}{{MRP}_{0}}.}}} & (90)\end{matrix}$For example, if MRP₀=$1,000,000, and c=$1000,α_(max)=$1000/$1,000,000=0.001. According to Equation 89, α_(t) wouldremain at this level in this example until n _(t)≧50,000 contracts,after which time it declines in proportion to 1/ n _(t). Equation 89indicates that α_(t) gradually grows in relation to 1/ n _(t) untilstabilizing at H/ n _(t).

Other formulas for variation of the price adjustment parameter are ofcourse possible and will also lead to convergence of Equations 85 and 86to the aggregate probability distribution of the market participants forthe various uncertain outcomes. One such possibility is to assessdirectly the degree to which market structure is in balance, in thesense that the ratios of prices to indicative payouts for the outcome ibeing purchased, v^(i) _(t)=P^(i) _(t)/[MRP_(t)/n^(i) _(t)], is wellreflected by the pricing probability, π^(i) _(t−1) used to compute P^(i)_(t). To the extent that v^(i) _(t)>π^(i) _(t−1), diagnostics of thistype would suggest that the pricing probability is too low relative tothe current anticipated payout, and its increase should be acceleratedthrough use of a larger value for α_(t). This formulation could beimplemented by setting, α_(t)=1/ n _(t) for v^(i) _(t)>π^(i) _(t−1); anddefining α_(t) as an increasing function of either v^(i) _(t)−π^(i)_(t−1), or v^(i) _(t)/π^(i) _(t−1), if v^(i) _(t)>π^(i) _(t−1).

Another class of formulations for variation of the price adjustmentparameter according to variations in market behavior relies on diagnosisof market imbalance through differences among outcomes of the numbers ofcontracts that have been bought, n^(i) _(t). Ideally, these should benearly equal across outcomes, and if so, α_(t)=1/ n _(t) is theappropriate value, as shown in the following appendix. Accordingly,indices of market imbalance can be computed to reflect nonconstancy ofthese contract numbers, n^(i) _(t). Examples of this class of indicesinclude the variance of the n^(i) _(t); the Gini coefficientcharacterizing these n^(i) _(t); or the discrepancy between thedistribution of the n^(i) _(t) and the ideal uniform distribution, asassessed by such indices as the cross-information or theKolmogorov-Smirnov statistic. Formulations in this class would beimplemented by defining α_(t) as an increasing function of any of theseindices.

22.4. An Idealized Example

FIG. 50 illustrates the capacity of the adaptive control algorithm inEquations 85 and 86 to converge to market participants' beliefs aboutthe outcome probabilities. Here the initial MRP is $1,000,000 and par is$1000, as in the examples above. Only five outcomes have been definedfor this hypothetical market, for the sake of clarity in the diagram.The initial probability assessment (according to which the market isinitially seeded) is that all five of these outcomes are equally likely,as indicated on the graph for MRP₀=$1,000,000.

The market participants do not agree that all the outcomes are equallylikely, and instead they invest money in proportion to the followingprobabilities for Outcomes 1 through 5, respectively: 0.30, 0.25, 0.20,0.15, and 0.10. Driven by this buying behavior, the pricingprobabilities promptly move toward these participant beliefs from theirinitial values, and the convergence is essentially complete afterapproximately $3,000,000 in participant investment in the market. Thisinvestment pattern continues until MRP=$15,000,000, at which time newinformation becomes available that indicates the probability for Outcome#1 is 0.6, and the probabilities for the remaining outcomes are all 0.1.From this point forward 60% of the new money coming into the market isinvested in contracts for Outcome #1, with the remainder split equallyamong the other four outcomes. Again the market pricing respondspromptly, and the pricing probabilities have essentially converged tothe participants' beliefs after an additional $3,000,000 has beeninvested.

22.5. Appendix

Derivation of the correct adjustment parameter α at economic equilibrium

This Appendix treats the case of a market in equilibrium, in which theextension of Equation 84 holds at a time t−1 for all outcomes i:

$\begin{matrix}{n_{t - 1}^{i} = {\frac{{MRP}_{t - 1}}{c\;{\exp\left\lbrack {{rj}/365} \right\rbrack}}.}} & (91)\end{matrix}$In most of the following, explicit indication of outcome i usingsuperscripts will be suppressed for notational simplicity.

Let v_(t)=P^(i) _(t)/[MRP_(t)/n^(i) _(t)] be the outcome probabilityimplied by the ratio of risk (price) to potential reward. The conditionexpressed in Equation A1 is consistent with this probability being equalto the pricing probability π^(i) _(t). We wish to increase the pricingprobability π_(t), using Equation 85, to match the increase in v_(t)resulting from the payout dilution for this outcome produced by thepurchase of one additional contract. Therefore,

$\begin{matrix}\begin{matrix}{\pi_{t} = {\pi_{t - 1} + {\alpha_{t}{\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)}}}} \\{= v_{t}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}{{MRP}_{t}/n_{t}}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}\left( {n_{t - 1} + 1} \right)}{{MRP}_{t - 1} + {\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}}} \\{= \frac{\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}\left( {n_{t - 1} + 1} \right)}{{n_{t - 1}c\;{\exp\left( {{rj}/365} \right)}} + {\pi_{t - 1}c\;{\exp\left( {{rj}/365} \right)}}}} \\{= \frac{\pi_{t - 1}\left( {n_{t - 1} + 1} \right)}{n_{t - 1} + \pi_{t - 1}}}\end{matrix} & (92)\end{matrix}$Here use has been made of the fact that, because of the equilibrium atstep t−1, MRP_(t−1)=n_(t−1) c exp(rj/365). Solving for the equilibriumadjustment parameter,

$\begin{matrix}\begin{matrix}{\alpha_{t} = \frac{\left\lbrack {\frac{\pi_{t - 1}\left( {n_{t - 1} + 1} \right)}{n_{t - 1} + \pi_{t - 1}} - \pi_{t - 1}} \right\rbrack}{\left\lbrack {\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)} \right\rbrack}} \\{= \frac{\pi_{t - 1}\left\lbrack {\left( {n_{t - 1} + 1} \right) - n_{t - 1} - \pi_{t - 1}} \right\rbrack}{{\pi_{t - 1}\left( {1 - \pi_{t - 1}} \right)}\left( {n_{t - 1} + \pi_{t - 1}} \right)}} \\{= \frac{1}{\left( {n_{t - 1} + \pi_{t - 1}} \right)}} \\{\approx {\frac{1}{n_{t - 1}}.}}\end{matrix} & (93)\end{matrix}$This approximation will be a very close one, because in realistic casesn_(t)>>π_(t).

Finally, because n^(i) _(t−1) is the same for each outcome i, theirarithmetic average is equal to any one of these, i.e.,

$\begin{matrix}{{{\overset{\_}{n}}_{t - 1} = {{\frac{1}{I}{\sum\limits_{i = 1}^{I}n_{t - 1}^{i}}} = n_{t - 1}}},} & (94)\end{matrix}$where I is the total number of outcomes. Thus Equation 93 becomes

$\begin{matrix}{\alpha_{t} = {\frac{1}{\left( {{\overset{\_}{n}}_{t - 1} + \pi_{t - 1}} \right)} \approx {\frac{1}{{\overset{\_}{n}}_{t - 1}}.}}} & (95)\end{matrix}$

22.6. ADDITIONAL EXAMPLES

The following gives examples of a First Landfall Market in which optionsor other investment units are made available for purchase at prices thatchange on an ongoing basis, once market activity commences. Preferably,before the first market activity, prices are preferably set usinghistorical climatological probabilities. Thereafter, as purchases aremade, prices are changed on an ongoing basis, preferably using equations85 and 86, above. Financial activity is also preferably carried outusing one or more computer systems and a graphical user interface,preferably, the one shown in FIGS. 24-49.

22.6.1 Example 1

In the first example, a computer implemented method for automaticallysetting prices of financial products in a financial activity having aplurality of possible outcomes, comprises the steps of:

receiving a first request from a participant terminal to purchase afinancial product for one of the possible outcomes, i; and

electronically computing a price for the requested financial product, inresponse to the first request, based at least in part on the followingfirst formulaP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome i    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1

If desired, the method could also include one or more steps forautomatically updating the prices for all other outcomes, other than theoutcome, i, based at least in part on the following second formulaP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing probability for        outcome k.    -   Kπ^(k) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1.

The method could further comprise the steps of receiving a secondrequest from a participant terminal to purchase a financial product forone of the possible outcomes; and delivering a price for the secondrequest to the user terminal after the price for the requested financialproduct is computed and the prices for all other outcomes are updated,in response to the first request.

If desired, setting an initial price for each outcome, could be providedprior to receiving an initial request, and a null outcome that is deemedto have occurred when none of the other outcomes have occurred could beadded to the method.

In one instance, the method could further comprise the step ofprocessing a request for multiple financial products for the sameoutcome by successively repeating the steps of calculating a price foreach financial product and updating prices for all other outcomes foreach financial product, taken one at a time.

In the method, the outcomes may be mutually exclusive, and/orcollectively exhaustive

If desired, the financial products may comprise contracts in a one-sidedmarket of buyer participants.

The first request from a participant terminal may be sent to a computerhaving memory with a data structure stored in memory, with the datastructure comprising the first and the second formula, above. Ifdesired, prices corresponding to the possible outcomes may also bestored in said memory.

22.6.2 Example 2

In a second example, a computer implemented method is provided forautomatically pricing financial products in a financial activity for aplurality of possible outcomes, consistent with the risks as perceivedat any given time by the activity participants, comprising the steps of:

receiving a first request from a participant terminal to purchase afinancial product for one of the possible outcomes, i;

electronically computing a price for the one possible outcome based atleast in part on the following first formulaP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1+α) _(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome I,    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and

updating all other prices based at least in part on the following secondformulaP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing probability for        outcome k.    -   Kπ^(i) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1.

The first request from a participant terminal may be sent to a computerhaving memory with a data structure stored in said memory, said datastructure comprising the first and the second formulae, and pricescorresponding to the possible outcomes may also be stored in saidmemory.

If desired, the step of setting an initial price for each outcome, priorto receiving an initial request could also be provided with or withoutan optional null outcome that is deemed to have occurred when none ofthe other outcomes have occurred.

The method could further comprise the steps of receiving a secondrequest from a participant terminal to purchase a financial product forone of the possible outcomes; and delivering a price for the secondrequest to the user terminal after the price for the requested financialproduct is computed and the prices for all other outcomes are updated,in response to the first request.

A request for multiple financial products for the same outcome could beprovided by successively repeating the steps of calculating a price foreach financial product and updating prices for all other outcomes foreach financial product, taken one at a time.

The outcomes may be mutually exclusive and/or collectively exhaustive

22.6.3 Example 3

In a third example, a computer implemented system is provided forautomatically setting prices of financial products in a financialactivity having a plurality of possible outcomes, comprising:

a network for receiving a first request from a participant terminal topurchase a financial product for one of the possible outcomes, i;

a computer having memory with a data structure stored in said memory,said data structure comprising the following first formulaP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome I,    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically computes a price for the requested        financial product, in response to the first request, based at        least in part on the first formula.

If desired, the data structure could further comprise the followingsecond formulaP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing probability for        outcome k.    -   Kπ^(k) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically automatically updates the prices for        all other outcomes, other than the outcome, i, based at least in        part on the second formula

Prices corresponding to the possible outcomes could be stored in saidmemory, and initial prices corresponding to the possible outcomes couldalso be stored in said memory, prior to receiving an initial request.

22.6.4 Example 4

In a fourth example, an article of manufacture includes a machinereadable medium for causing a computer system to automatically setprices of financial products in a financial activity having a pluralityof possible outcomes, comprising:

an input for receiving a first request from a participant terminal topurchase a financial product for one of the possible outcomes; i; and

a data structure comprising the following first formulaP ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))]

where

-   -   i is the outcome requested by the participant,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(i) _(t) is the price for the purchase requested by the        participant    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term annualized        interest rate,    -   j is the relative Julian date since the financial activity was        started,    -   π^(i) _(t−1) is the last previously calculated pricing        probability for outcome I,    -   Kπ^(i) _(t−1) is the last previously calculated price for        outcome i, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically computes a price for the requested        financial product, in response to the first request, based at        least in part on the first formula.

The machine readable medium could further comprise the following secondformulaP ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠i

where

-   -   k, where k≠i represents the set of all other outcomes,    -   t is a time index counter designating the current transaction,    -   t−1 is a time index counter designating the last previous        transaction,    -   P^(k) _(t) where k≠i is the set of all other prices, updated to        take into account the purchase requested by the participant,    -   π^(k) _(t−1) is the latest calculated pricing probability for        outcome k.    -   Kπ^(k) _(t−1) is the latest calculated price for outcome k.    -   K=c exp [rj/365],    -   c is a scaling constant,    -   r is a constant proportional to the short term cost of money,    -   j is the relative Julian date since the financial activity was        started, and    -   α_(t) is a price adjustment parameter, having a value between 0        and 0.1; and    -   the computer electronically automatically updates the prices for        all other outcomes, other than the outcome, i, based at least in        part on the second formula

The machine readable medium could further comprise instructions for thecomputer to store prices corresponding to the possible outcomes in saidmemory.

-   -   Instructions could also be provided for the computer to store        initial prices corresponding to the possible outcomes, prior to        receiving an initial request        XXIII. Preferred Characteristics

As noted above, a number of different alternatives and variations inconducting financial activities are possible. The following discussesrepresentative alternatives and variations which are preferred, but notnecessarily required. Although the following exemplary preferredcharacteristics may, generally speaking, be compatible with one another,it is possible that any number of these characteristics could be madeinconsistent with, or mutually exclusive of other characteristics. Theseexemplary characteristics include:

1. Variability factors affecting at least one of said investment priceand said distribution/payout. It is generally preferred that variabilityfactors include, at least a consideration of the time interval betweeninvestment and event occurrence and a defined probability of predictedoutcome, preferably set at the time of investment. Other variabilityfactors may also be incorporated.

2. Prices charged to participants for their chosen investmentspreferably continually change due, for example, to the variabilityfactors at play at a given time. In general, it is preferred that therebe no elimination of price changes to shorten processing delays, or forother reasons.

3. Prices at any given time for any predicted outcome are preferablymade to be the same for all participants.

4. Payouts to successful, qualifying participants are preferably madeaccording to the same set of rules which apply to all participants.Generally speaking, it is preferred that no rewards be given forpreferred participants.

5. All winners (qualifying participants) of the financial activity sharethe pot. That is, it is generally preferred that there are no oddsmultiplying a participant's investment. Also, it is generally preferredthat payouts are not made from the provider's personal account—unlikethe “House” of certain gambling activities which pays out winning betsfrom its own account.

6. Provider does not engage as a participant. For example, it isgenerally preferred it that there be no hedging where, for example,there may be excessive bidding.

7. Participants do not compete against the “house” i.e. the provider.

8. In certain instances, it may be preferable to limit financialactivity to only the United States and its territories and possessions.

9. Financial activity encompasses a single event or type of event over agiven “season”.

10. It is generally preferred that the participants be able toobjectively and independently observe the events for themselves, as theyunfold.

11. It is generally preferred that, apart from financial responsibility,a participant's investment be “accepted” only in terms of data format.Optionally, acceptance can be related to an optional investment cap. Itis generally preferred that there be no extra qualification for eachinvestment occurrence.

12. It is generally preferred that financial activity be carried out asmuch as possible in real-time, and that this be made possible by virtueof rules definitions, especially definitions of events and eventoutcomes which occur in a well defined environment/system. It isgenerally preferred that the selection of events for the financialactivity be limited to an absolute minimum—i.e. a single event.

13. It is generally preferred that the financial activity be restrictedto tropical weather events, and in one instance, only to hurricaneevents.

14. It is generally preferred that the financial activity be furtherrestricted to events comprising at least one of said landfall and saidland track. It may also be preferable in certain instances to limitfinancial activity to hurricanes having a certain category strength atsome defined time during the ongoing financial activity.

15. It is preferable, in certain instances, that multiple stages ofprobability assessment be applied to at least one of said investment andsaid payout.

16. It is preferable in certain instances that the external objectiveindependent information source be limited to the National HurricaneCenter (NHC), and optionally to its subsidiary and/or related agencies(e.g. TPC).

XXIV. Other Preferences

In addition to the above preferred characteristics, it is generallypreferred that the graphical user interface be used to conveyeducational information to participants, on an ongoing, developingbasis. For example, it is generally preferred that the participants beprovided with a rich source of information to inspire further interestin the financial activity and the skills which it sharpens. For example,it is generally preferred that live feeds of various external objectiveindependent information sources be relayed to the participants on anongoing basis. For example, storm locations and changing intensities,along with their projected path can be displayed on a flat map. Thistype of data can be obtained from the National Hurricane Center, forexample. Other competing predictions from other independent sourcescould also be displayed, preferably in different colors on a common map.

As storms advance and locations change, it is preferred thatcalculations of updated current pricing and returns be displayed on anongoing basis, with any necessary qualifying assumptions being madeavailable to the participants.

Also, it is generally preferred that the current dollars invested forparticular geographical areas be displayed along with historical odds orprobabilities for the geographical area, thus making it possible forparticipants and interested observers to determine what the payout willbe under the specified conditions, if the storm should develop aspredicted. For example, payoffs for a given geographical area canreflect different times of investment, different total amounts of moneyinvested in the overall financial activity, different severity levels orother characteristics of the storm and the probability that the stormwill hit the area of interest, based upon historical data and/or nearterm predictions. It is also generally preferred that the participantsand interested observers be able to access a data window showing thecurrent total of pools invested, and a profile of geographical areaswith the number of investments being made for that local area.

In general, it is preferred that the financial activity have aneducational study component to sharpen a participant's knowledge andskills useful for improving their financial position. This knowledge andskill level will also help the participants cope with the reality ofbeing subjected to potentially harmful storm activity. It is alsogenerally preferred that the financial activity display helpfulinformation to participants, such as checklists of items needed toprepare for an oncoming threatening event. These checklists can helporganize participant's activities in the decreasing preparation timeavailable. If desired, checklist information can be solicited fromparticipants and posted in a public viewing area. In another example, adisplay area can also be provided showing constantly updated damageestimates, threats to life and similar public safety-relatedinformation.

Although exemplary implementations of the invention have been depictedand described in detail herein, it will be apparent to those skilled inthe relevant art that various modifications, additions, substitutions,and the like can be made without departing from the spirit of theinvention and these are therefore considered to be within the scope ofthe invention as defined in the following claims. For example, variouscommunication functions can be grouped into one or more communicationunits to perform one or more communication tasks. Any of the examplesand alternatives provided herein could be combined in whole or in partwith one another, as may be desired, and can be combined with one ormore features set forth herein.

1. A computer implemented method for automatically setting prices offinancial products in a financial activity having a plurality ofpossible outcomes, comprising the steps of: receiving a first requestfrom a first participant terminal of a plurality of participantterminals to purchase a financial product for one outcome of thepossible outcomes, i; and electronically computing a price for thefinancial product, in response to the first request, based at least inpart on the following first formula:P ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))] where i designates the one outcome requested by the firstparticipant, t is a time index counter designating the current purchaserequested by the first participant, t−1 is a time index counterdesignating a last previous transaction purchase, P^(i) _(t) is theprice for the financial product, K=c exp[rj/365], c is a money amountused as a scaling constant, r is a constant proportional to a short termannualized interest rate, j is a number of days since the financialactivity was started, π^(i) _(t) is the current pricing probability foroutcome i, π^(i) _(t−1) is the last previously computed pricingprobability for outcome i, α_(t) is a price adjustment parameter, havinga value greater than 0 and less than 0.1, and delivering the price forthe financial product of the first request to the first participantterminal.
 2. The method according to claim 1, further comprising thestep of: automatically updating prices for all other outcomes of thepossible outcomes based at least in part on the following secondformula:P ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠iwhere k, where k≠i, represents a set of said all other outcomes of thepossible outcomes, t is the time index counter designating the currentpurchase requested by the first participant, t−1 is the time indexcounter designating the last previous purchase, P^(k) _(t) where k≠i, isa set of all other prices, updated to take into account the currentpurchase requested by the first participant, π^(k) _(t) is the currentpricing probability for outcome k, π^(k) _(t−1) is the latest computedpricing probability for outcome k, π^(i) _(t−1) is the last previouslycomputed pricing probability for outcome i, K=c exp[rj/365], c is themoney amount used as the scaling constant, r is a constant proportionalto the short term annualized interest rate, j is the number of dayssince the financial activity was started, and α_(t) is a priceadjustment parameter, having a value greater than 0 and less than 0.1.3. The method according to claim 2, further comprising the steps of:receiving a second request from a second participant terminal of theplurality of participant terminals to purchase a financial product foranother outcome of the possible outcomes; and delivering a price for thefinancial product of the second request to the second participantterminal after the prices for said all other outcomes of the possibleoutcomes are automatically updated and the price for the financialproduct of the second requests is computed based at least in part on thefirst formula.
 4. The method according to claim 1, further comprisingthe step of: setting an initial price for each outcome of the possibleoutcomes, prior to receiving any request.
 5. The method according toclaim 1, further comprising the step of: adding a null outcome as anoutcome of the possible outcomes, wherein the null outcome is deemed tohave occurred when no other possible outcomes have occurred.
 6. Themethod according to claim 2, further comprising the step of: processinga second request to purchase multiple financial products for a sameoutcome of the possible outcomes by successively repeating, for eachfinancial product of the requested multiple financial products, thesteps of (a) electronically computing a price for said each financialproduct based at least in part on the first formula and (b)automatically updating prices for all other outcomes of the possibleoutcomes based at least in part on the second formula.
 7. The methodaccording to claim 1, wherein the possible outcomes are mutuallyexclusive.
 8. The method according to claim 1, wherein the possibleoutcomes are collectively exhaustive.
 9. The method according to claim1, wherein the possible outcomes are mutually exclusive and collectivelyexhaustive.
 10. The method according to claim 1, wherein the financialproduct comprises a contract in a one-sided market of buyerparticipants.
 11. The method according to claim 2, wherein the firstrequest from the first participant terminal is sent to a computer havingmemory with a data structure stored in said memory, said data structurecomprising the first formula and the second formula.
 12. The methodaccording to claim 11, wherein the computed and updated pricescorresponding to the possible outcomes are stored in said memory.
 13. Acomputer implemented method for automatically pricing financial productsin a financial activity for a plurality of possible outcomes, consistentwith the risks as perceived at any given time by the activityparticipants, comprising the steps of: receiving a first request from afirst participant terminal of a plurality of participant terminals topurchase a financial product for one outcome of the possible outcomes,i; electronically computing a price for the financial product based atleast in part on the following first formula:P ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))] where i designates the one outcome requested by the firstparticipant, t is a time index counter designating the current purchaserequested by the first participant, t−1 is a time index counterdesignating a last previous purchase, P^(i) _(t) is the price for thefinancial product, K=c exp[rj/365], c is a money amount used as ascaling constant, r is a constant proportional to a short termannualized interest rate, j is a number of days since the financialactivity was started, π^(i) _(t) is the current pricing probability foroutcome i, π^(i) _(t−1) is the last previously computed pricingprobability for outcome i, α_(t) is a price adjustment parameter, havinga value greater than 0 and less than 0.1; automatically updating pricesfor all other outcomes of the possible outcomes based at least in parton the following second formula:P ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠iwhere k, where k≠i, represents a set of said all other outcomes of thepossible outcomes, t is the time index counter designating the currentpurchase requested by the first participant, t−1 is the time indexcounter designating the last previous purchase, P^(k) _(t), where k≠i,is a set of all other prices, updated to take into account the currentpurchase requested by the first participant, π^(k) _(t) is the currentpricing probability for outcome k, π^(k) _(t−1) is the latest computedpricing probability for outcome k, π^(i) _(t−1) is the last previouslycomputed pricing probability for outcome i, K=c exp[rj/365], c is themoney amount used as the scaling constant, r is a constant proportionalto the short term annualized interest rate, j is the number of dayssince the financial activity was started, and α_(t) is a priceadjustment parameter, having a value between greater than 0 and lessthan 0.1 and delivering the price for the financial product of the firstrequest to the first participant terminal.
 14. The method according toclaim 13, wherein the first request from the first participant terminalis sent to a computer having memory with a data structure stored in saidmemory, said data structure comprising the first formula and the secondformula.
 15. The method according to claim 13, wherein the computed andupdated prices corresponding to the possible outcomes are stored in saidmemory.
 16. The method according to claim 13, further comprising thestep of: setting an initial price for each outcome of the possibleoutcomes, prior to receiving any request.
 17. The method according toclaim 13, further comprising the step of: adding a null outcome as anoutcome of the possible outcomes, wherein the null outcome is deemed tohave occurred when no other possible outcomes have occurred.
 18. Themethod according to claim 13, further comprising the steps of: receivinga second request from a second participant terminal of the plurality ofparticipant terminals to purchase a financial product for anotheroutcome of the possible outcomes; and delivering a price for thefinancial product of the second request to the second participantterminal after the prices for said all other outcomes of the possibleoutcomes are automatically updated and the price for the financialproduct of the second request is computed based at least in part on thefirst formula.
 19. The method according to claim 13, further comprisingthe step of: processing a second request to purchase multiple financialproducts for a same outcome of the possible outcomes by successivelyrepeating, for each financial product of the requested multiplefinancial products, the steps of (a) electronically computing a pricefor said each financial product based at least in part on the firstformula and (b) automatically updating prices for all other outcomes ofthe possible outcomes based at least in part on the second formula. 20.The method according to claim 13, wherein the possible outcomes aremutually exclusive and collectively exhaustive.
 21. A computerimplemented system for automatically setting prices of financialproducts in a financial activity having a plurality of possibleoutcomes, comprising: a network configured to receive a first requestfrom a first participant terminal of a plurality of participantterminals to purchase a financial product for one outcome of thepossible outcomes, i; a computer having memory with a data structurestored in said memory, said data structure comprising the followingfirst formula:P ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)(1−π^(i) _(t−1))] where idesignates the one outcome requested by the first participant, t is atime index counter designating the current purchase requested by thefirst participant, t−1 is a time index counter designating a lastprevious purchase, ^(i) _(t) is the price for the financial product, K=cexp[rj/365], c is a money amount used as a scaling constant, r is aconstant proportional to a short term annualized interest rate, j is anumber of days since the financial activity was started, π^(i) _(t) isthe current pricing probability for outcome i, π^(i) _(t−1) is the lastpreviously computed pricing probability for outcome i, α_(t) is a priceadjustment parameter, having a value greater than 0 and less than 0.1;and wherein the computer executes instructions to electronically computea price for the financial product, in response to the first request,based at least in part on the first formula and to deliver the price forthe financial product of the first request to the first participantterminal.
 22. The system according to claim 21, wherein said datastructure further comprises the following second formula:P ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠iwhere k, where k≠i, represents a set of all other outcomes of thepossible outcomes, t is the time index counter designating the currentpurchase requested by the first participant, t−1 is the time indexcounter designating the last previous purchase, P^(k) _(t), where k≠i,is a set of all other prices, updated to take into account the currentpurchase requested by the first participant, π^(k) _(t) is the currentpricing probability for outcome k, π^(k) _(t−1) is the latest computedpricing probability for outcome k, π^(i) _(t−1) is the last previouslycomputed pricing probability for outcome i, K=c exp[rj/365], c is themoney amount used as the scaling constant, r is a constant proportionalto the short term annualized interest rate, j is the number of dayssince the financial activity was started, and α_(t) is a priceadjustment parameter, having a value greater than 0 and less than 0.1;and wherein the computer executes further instructions to automaticallyupdate prices for said all other outcomes of the possible outcomes basedat least in part on the second formula.
 23. The system according toclaim 22, wherein the computed and updated prices corresponding to thepossible outcomes are stored in said memory.
 24. The system according toclaim 22, wherein initial prices for each outcome of the possibleoutcomes are stored in said memory, prior to receiving any request. 25.An article of manufacture including a non-transitory machine readablemedium for causing a computer system to automatically set prices offinancial products in a financial activity having a plurality ofpossible outcomes, the non-transitory machine readable mediumcomprising: instructions to receive a first request from a firstparticipant terminal of a plurality of participant terminals to purchasea financial product for one outcome of the possible outcomes, i; a datastructure comprising the following first formula:P ^(i) _(t) =Kπ ^(i) _(t) =K[π ^(i) _(t−1)+α_(t)π^(i) _(t−1)(1−π^(i)_(t−1))] where i designates the one outcome requested by the firstparticipant, t is a time index counter designating the current purchaserequested by the first participant, t−1 is a time index counterdesignating a last previous purchase, P^(i) _(t) is a price for thefinancial product, K=c exp[rj/365], c is a money amount used as ascaling constant, r is a constant proportional to a short termannualized interest rate, j is a number of days since the financialactivity was started, π^(i) _(t) is current pricing probability foroutcome i, π^(i) _(t−1) is the last previously computed pricingprobability for outcome i, α_(t) is a price adjustment parameter, havinga value greater than 0 and less than 0.1; and instructions toelectronically compute the price for the financial product, in responseto the first request, based at least in part on the first formula and todeliver the price for the financial product of the first request to thefirst participant terminal.
 26. The article of manufacture according toclaim 25, wherein said data structure further comprises the followingsecond formula:P ^(k) _(t) =Kπ ^(k) _(t) =K[π ^(k) _(t−1)(1−α_(t)π^(i) _(t−1))], k≠iwhere k, where k≠i, represents a set of all other outcomes of thepossible outcomes, t is the time index counter designating the currentpurchase requested by the first participant, t−1 is the time indexcounter designating the last previous purchase, P^(k) _(t), where k≠i,is a set of all other prices, updated to take into account the currentpurchase requested by the participant, π^(k) _(t) is the current pricingprobability for outcome k, π^(k) _(t−1) is the latest computed pricingprobability for outcome k, π^(i) _(t−1) is the last previously computedpricing probability for outcome i, K=c exp[rj/365], c is the moneyamount used as the scaling constant, r is a constant proportional to theshort term annualized interest rate, j is the number of days since thefinancial activity was started, α_(t) is a price adjustment parameter,having a value greater than 0 and less than 0.1; and wherein saidnon-transitory machine readable medium further comprises instructions toautomatically update prices for said all other outcomes of the possibleoutcomes based at least in part on the second formula.
 27. The articleof manufacture according to claim 26, wherein the non-transitory machinereadable medium further comprises instructions to store the computed andupdated prices corresponding to the of possible outcomes.
 28. Thearticle of manufacture according to claim 27, wherein the non-transitorymachine readable medium further comprises instructions to store initialprices for each outcome of the possible outcomes, prior to receiving anyrequest.